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The Matrix of a Linear Transformation
Linear AlgebraMATH 2076
Section 4.7 The Matrix of an LT 27 March 2017 1 / 7
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The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e.,
there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
![Page 3: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/3.jpg)
The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x .
In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
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The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
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The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?
Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
![Page 6: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/6.jpg)
The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors.
Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
![Page 7: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/7.jpg)
The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
![Page 8: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/8.jpg)
The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm.
Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
![Page 9: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/9.jpg)
The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and
given ~w in W, we get[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
![Page 10: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/10.jpg)
The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
![Page 11: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/11.jpg)
The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v).
Bylinearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
![Page 12: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/12.jpg)
The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation.
So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
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The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
![Page 14: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/14.jpg)
The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A =
~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B. We call M the matrix for T relative to B and A
and we write[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
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The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:
[T (~v)
]A =
[~w]A = ~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B. We call M the matrix for T relative to B and A
and we write[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
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The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:[
T (~v)]A =
[~w]A = ~y = M~x
= M[~v]B;
i.e.,[T (~v)
]A = M
[~v]B. We call M the matrix for T relative to B and A
and we write[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
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The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:[
T (~v)]A =
[~w]A = ~y = M~x = M
[~v]B;
i.e.,[T (~v)
]A = M
[~v]B. We call M the matrix for T relative to B and A
and we write[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
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The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:[
T (~v)]A =
[~w]A = ~y = M~x = M
[~v]B;
i.e.,[T (~v)
]A = M
[~v]B.
We call M the matrix for T relative to B and Aand we write
[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
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The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:[
T (~v)]A =
[~w]A = ~y = M~x = M
[~v]B;
i.e.,[T (~v)
]A = M
[~v]B. We call M the matrix for T relative to B and A
and
we write[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
![Page 20: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/20.jpg)
The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:[
T (~v)]A =
[~w]A = ~y = M~x = M
[~v]B;
i.e.,[T (~v)
]A = M
[~v]B. We call M the matrix for T relative to B and A
and we write[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
![Page 21: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/21.jpg)
The Matrix of a Linear Transformation
Recall that every LT Rn T−→ Rm is a matrix transformation; i.e., there is anm × n matrix A so that T (~x) = A~x . In fact, Colj(A) = T (~ej).
Suppose V T−→W is a LT. Can we view T as a matrix transformation?Yes, if we use coordinate vectors. Let B,A be bases for V,W resp.
Consider the coordinate maps V [·]B−−→ Rn and W [·]A−−→ Rm. Given ~v in V,we get
[~v]B in Rn, and given ~w in W, we get
[~w]A in Rm.
Consider ~x =[~v]B and ~y =
[~w]A where ~v is in V and ~w = T (~v). By
linearity props of coord vectors, the map ~x 7→ ~y (from Rn to Rm) is a lineartransformation. So ~x 7→ ~y is given by multiplication by some matrix M:[
T (~v)]A =
[~w]A = ~y = M~x = M
[~v]B;
i.e.,[T (~v)
]A = M
[~v]B. We call M the matrix for T relative to B and A
and we write[T]AB= M, so
[T (~v)
]A =
[T]AB[~v]B .
Section 4.7 The Matrix of an LT 27 March 2017 2 / 7
![Page 22: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/22.jpg)
Picture for the matrix for T relative to B and A
We have a linear transformation V T−→W and bases B,A for V,W resp.
Consider the B and A coord maps V→ Rn
and W→ Rm. Given ~v in V, ~w in W, let~x =
[~v]B and ~y =
[~w]A.
Consider the LT Rn → Rm given by ~x 7→ ~y .This is a matrix transformation, and
[T (~v)
]A =
[~w]A =
~y = M~x
= A[~v]B
where M =[T]AB.
V WT
Rn Rm
[·]B [·]A
~v ~w
~x ~y
Section 4.7 The Matrix of an LT 27 March 2017 3 / 7
![Page 23: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/23.jpg)
Picture for the matrix for T relative to B and A
We have a linear transformation V T−→W and bases B,A for V,W resp.
Consider the B and A coord maps V→ Rn
and W→ Rm.
Given ~v in V, ~w in W, let~x =
[~v]B and ~y =
[~w]A.
Consider the LT Rn → Rm given by ~x 7→ ~y .This is a matrix transformation, and
[T (~v)
]A =
[~w]A =
~y = M~x
= A[~v]B
where M =[T]AB.
V WT
Rn Rm
[·]B [·]A
~v ~w
~x ~y
Section 4.7 The Matrix of an LT 27 March 2017 3 / 7
![Page 24: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/24.jpg)
Picture for the matrix for T relative to B and A
We have a linear transformation V T−→W and bases B,A for V,W resp.
Consider the B and A coord maps V→ Rn
and W→ Rm. Given ~v in V, ~w in W, let
~x =[~v]B and ~y =
[~w]A.
Consider the LT Rn → Rm given by ~x 7→ ~y .This is a matrix transformation, and
[T (~v)
]A =
[~w]A =
~y = M~x
= A[~v]B
where M =[T]AB.
V WT
Rn Rm
[·]B [·]A
~v ~w
~x ~y
Section 4.7 The Matrix of an LT 27 March 2017 3 / 7
![Page 25: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/25.jpg)
Picture for the matrix for T relative to B and A
We have a linear transformation V T−→W and bases B,A for V,W resp.
Consider the B and A coord maps V→ Rn
and W→ Rm. Given ~v in V, ~w in W, let~x =
[~v]B and ~y =
[~w]A.
Consider the LT Rn → Rm given by ~x 7→ ~y .This is a matrix transformation, and
[T (~v)
]A =
[~w]A =
~y = M~x
= A[~v]B
where M =[T]AB.
V WT
Rn Rm
[·]B [·]A
~v ~w
~x ~y
Section 4.7 The Matrix of an LT 27 March 2017 3 / 7
![Page 26: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/26.jpg)
Picture for the matrix for T relative to B and A
We have a linear transformation V T−→W and bases B,A for V,W resp.
Consider the B and A coord maps V→ Rn
and W→ Rm. Given ~v in V, ~w in W, let~x =
[~v]B and ~y =
[~w]A.
Consider the LT Rn → Rm given by ~x 7→ ~y .
This is a matrix transformation, and
[T (~v)
]A =
[~w]A =
~y = M~x
= A[~v]B
where M =[T]AB.
V WT
Rn Rm
[·]B [·]A
~v ~w
~x ~y
Section 4.7 The Matrix of an LT 27 March 2017 3 / 7
![Page 27: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/27.jpg)
Picture for the matrix for T relative to B and A
We have a linear transformation V T−→W and bases B,A for V,W resp.
Consider the B and A coord maps V→ Rn
and W→ Rm. Given ~v in V, ~w in W, let~x =
[~v]B and ~y =
[~w]A.
Consider the LT Rn → Rm given by ~x 7→ ~y .This is a matrix transformation, and
[T (~v)
]A =
[~w]A =
~y = M~x
= A[~v]B
where M =[T]AB.
V WT
Rn Rm
[·]B [·]A
~v ~w
~x ~y
Section 4.7 The Matrix of an LT 27 March 2017 3 / 7
![Page 28: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/28.jpg)
Picture for the matrix for T relative to B and A
We have a linear transformation V T−→W and bases B,A for V,W resp.
Consider the B and A coord maps V→ Rn
and W→ Rm. Given ~v in V, ~w in W, let~x =
[~v]B and ~y =
[~w]A.
Consider the LT Rn → Rm given by ~x 7→ ~y .This is a matrix transformation, and
[T (~v)
]A =
[~w]A =
~y = M~x
= A[~v]B
where M =[T]AB.
V WT
Rn Rm
[·]B [·]A
~v ~w
~x ~y
Section 4.7 The Matrix of an LT 27 March 2017 3 / 7
![Page 29: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/29.jpg)
Picture for the matrix for T relative to B and A
We have a linear transformation V T−→W and bases B,A for V,W resp.
Consider the B and A coord maps V→ Rn
and W→ Rm. Given ~v in V, ~w in W, let~x =
[~v]B and ~y =
[~w]A.
Consider the LT Rn → Rm given by ~x 7→ ~y .This is a matrix transformation, and
[T (~v)
]A =
[~w]A = ~y = M~x
= A[~v]B
where M =[T]AB.
V WT
Rn Rm
[·]B [·]A
~v ~w
~x ~y
Section 4.7 The Matrix of an LT 27 March 2017 3 / 7
![Page 30: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/30.jpg)
Picture for the matrix for T relative to B and A
We have a linear transformation V T−→W and bases B,A for V,W resp.
Consider the B and A coord maps V→ Rn
and W→ Rm. Given ~v in V, ~w in W, let~x =
[~v]B and ~y =
[~w]A.
Consider the LT Rn → Rm given by ~x 7→ ~y .This is a matrix transformation, and[T (~v)
]A =
[~w]A = ~y = M~x
= A[~v]B
where M =[T]AB.
V WT
Rn Rm
[·]B [·]A
~v ~w
~x ~y
Section 4.7 The Matrix of an LT 27 March 2017 3 / 7
![Page 31: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/31.jpg)
Picture for the matrix for T relative to B and A
We have a linear transformation V T−→W and bases B,A for V,W resp.
Consider the B and A coord maps V→ Rn
and W→ Rm. Given ~v in V, ~w in W, let~x =
[~v]B and ~y =
[~w]A.
Consider the LT Rn → Rm given by ~x 7→ ~y .This is a matrix transformation, and[T (~v)
]A =
[~w]A = ~y = M~x = A
[~v]B
where M =[T]AB.
V WT
Rn Rm
[·]B [·]A
~v ~w
~x ~y
Section 4.7 The Matrix of an LT 27 March 2017 3 / 7
![Page 32: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/32.jpg)
Picture for the matrix for T relative to B and A
We have a linear transformation V T−→W and bases B,A for V,W resp.
Consider the B and A coord maps V→ Rn
and W→ Rm. Given ~v in V, ~w in W, let~x =
[~v]B and ~y =
[~w]A.
Consider the LT Rn → Rm given by ~x 7→ ~y .This is a matrix transformation, and[T (~v)
]A =
[~w]A = ~y = M~x = A
[~v]B
where M =[T]AB.
V WT
Rn Rm
[·]B [·]A
~v ~w
~x ~y
Section 4.7 The Matrix of an LT 27 March 2017 3 / 7
![Page 33: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/33.jpg)
Finding the Matrix for a Linear Transformation
Suppose V T−→W is a linear transformation and B,A are bases for V,Wresp. Then
[T (~v)
]A =
[T]AB[~v]B where
[T]AB is the matrix for T
relative to B and A.
To find[T]AB, we need to know B. Suppose B = {~b1, ~b2, . . . , ~bn}. Then
Colj
([T]AB
)=[T (~bj)
]A .
When V = Rn,W = Rm and B,A are the standard bases, this is the usualformula for the standard matrix for T .
You can remember this as[T]AB=
[T (B)
]A, but this abuses notation!
Section 4.7 The Matrix of an LT 27 March 2017 4 / 7
![Page 34: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/34.jpg)
Finding the Matrix for a Linear Transformation
Suppose V T−→W is a linear transformation and B,A are bases for V,Wresp. Then
[T (~v)
]A =
[T]AB[~v]B
where[T]AB is the matrix for T
relative to B and A.
To find[T]AB, we need to know B. Suppose B = {~b1, ~b2, . . . , ~bn}. Then
Colj
([T]AB
)=[T (~bj)
]A .
When V = Rn,W = Rm and B,A are the standard bases, this is the usualformula for the standard matrix for T .
You can remember this as[T]AB=
[T (B)
]A, but this abuses notation!
Section 4.7 The Matrix of an LT 27 March 2017 4 / 7
![Page 35: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/35.jpg)
Finding the Matrix for a Linear Transformation
Suppose V T−→W is a linear transformation and B,A are bases for V,Wresp. Then
[T (~v)
]A =
[T]AB[~v]B where
[T]AB is the matrix for T
relative to B and A.
To find[T]AB, we need to know B. Suppose B = {~b1, ~b2, . . . , ~bn}. Then
Colj
([T]AB
)=[T (~bj)
]A .
When V = Rn,W = Rm and B,A are the standard bases, this is the usualformula for the standard matrix for T .
You can remember this as[T]AB=
[T (B)
]A, but this abuses notation!
Section 4.7 The Matrix of an LT 27 March 2017 4 / 7
![Page 36: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/36.jpg)
Finding the Matrix for a Linear Transformation
Suppose V T−→W is a linear transformation and B,A are bases for V,Wresp. Then
[T (~v)
]A =
[T]AB[~v]B where
[T]AB is the matrix for T
relative to B and A.
To find[T]AB, we need to know B.
Suppose B = {~b1, ~b2, . . . , ~bn}. Then
Colj
([T]AB
)=[T (~bj)
]A .
When V = Rn,W = Rm and B,A are the standard bases, this is the usualformula for the standard matrix for T .
You can remember this as[T]AB=
[T (B)
]A, but this abuses notation!
Section 4.7 The Matrix of an LT 27 March 2017 4 / 7
![Page 37: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/37.jpg)
Finding the Matrix for a Linear Transformation
Suppose V T−→W is a linear transformation and B,A are bases for V,Wresp. Then
[T (~v)
]A =
[T]AB[~v]B where
[T]AB is the matrix for T
relative to B and A.
To find[T]AB, we need to know B. Suppose B = {~b1, ~b2, . . . , ~bn}. Then
Colj
([T]AB
)=[T (~bj)
]A .
When V = Rn,W = Rm and B,A are the standard bases, this is the usualformula for the standard matrix for T .
You can remember this as[T]AB=
[T (B)
]A, but this abuses notation!
Section 4.7 The Matrix of an LT 27 March 2017 4 / 7
![Page 38: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/38.jpg)
Finding the Matrix for a Linear Transformation
Suppose V T−→W is a linear transformation and B,A are bases for V,Wresp. Then
[T (~v)
]A =
[T]AB[~v]B where
[T]AB is the matrix for T
relative to B and A.
To find[T]AB, we need to know B. Suppose B = {~b1, ~b2, . . . , ~bn}. Then
Colj
([T]AB
)=[T (~bj)
]A .
When V = Rn,W = Rm and B,A are the standard bases, this is the usualformula for the standard matrix for T .
You can remember this as[T]AB=
[T (B)
]A, but this abuses notation!
Section 4.7 The Matrix of an LT 27 March 2017 4 / 7
![Page 39: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/39.jpg)
Finding the Matrix for a Linear Transformation
Suppose V T−→W is a linear transformation and B,A are bases for V,Wresp. Then
[T (~v)
]A =
[T]AB[~v]B where
[T]AB is the matrix for T
relative to B and A.
To find[T]AB, we need to know B. Suppose B = {~b1, ~b2, . . . , ~bn}. Then
Colj
([T]AB
)=[T (~bj)
]A .
When V = Rn,W = Rm and B,A are the standard bases, this is the usualformula for the standard matrix for T .
You can remember this as[T]AB=
[T (B)
]A, but this abuses notation!
Section 4.7 The Matrix of an LT 27 March 2017 4 / 7
![Page 40: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/40.jpg)
Finding the Matrix for a Linear Transformation
Suppose V T−→W is a linear transformation and B,A are bases for V,Wresp. Then
[T (~v)
]A =
[T]AB[~v]B where
[T]AB is the matrix for T
relative to B and A.
To find[T]AB, we need to know B. Suppose B = {~b1, ~b2, . . . , ~bn}. Then
Colj
([T]AB
)=[T (~bj)
]A .
When V = Rn,W = Rm and B,A are the standard bases, this is the usualformula for the standard matrix for T .
You can remember this as[T]AB=
[T (B)
]A, but this abuses notation!
Section 4.7 The Matrix of an LT 27 March 2017 4 / 7
![Page 41: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/41.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn.
Here we write[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B =
P−1~y
=
P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 42: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/42.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B,
and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B =
P−1~y
=
P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 43: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/43.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B =
P−1~y
=
P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 44: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/44.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn.
Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B =
P−1~y
=
P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 45: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/45.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at
[~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B =
P−1~y
=
P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 46: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/46.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B =
P−1~y
=
P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 47: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/47.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B =
P−1~y
=
P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 48: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/48.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that
[T]B[~x]B =
[T (~x)
]B =
[~y]B =
P−1~y
=
P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 49: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/49.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B =
P−1~y
=
P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 50: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/50.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B =
P−1~y
=
P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 51: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/51.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B =
P−1~y
=
P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 52: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/52.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B = P−1~y
=
P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 53: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/53.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B = P−1~y
= P−1A~x =
P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 54: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/54.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B = P−1~y
= P−1A~x = P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 55: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/55.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B = P−1~y
= P−1A~x = P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 56: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/56.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B = P−1~y
= P−1A~x = P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1.
Thus A =[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 57: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/57.jpg)
The B-Matrix for a Linear Transformation
Suppose V = Rn = W and B = A, so Rn T−→ Rn is a linear transformationand B is some basis for Rn. Here we write
[T]B=[T]BB for the matrix for
T relative to B and B, and call this the the B-matrix for T .
Consider the B coord map Rn → Rn. Given ~xand ~y = T (~x) = A~x , look at[
~x]B and
[~y]B =
[A~x]B.
Recall that ~x = P[~x]B where P = PEB =
[B].
So[~y]B = P−1~y and we see that[
T]B[~x]B =
[T (~x)
]B =
[~y]B = P−1~y
= P−1A~x = P−1AP[~x]B.
Rn RnT
Rn Rn
[·]B [·]B
~x ~y = A~x
[~x]B
[~y]B
It follows that the B-matrix for T is given by[T]B = P−1AP, so
A = P[T]BP−1. Thus A =
[T]E and
[T]B are similar matrices!
Section 4.7 The Matrix of an LT 27 March 2017 5 / 7
![Page 58: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/58.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix.
Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=
[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 59: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/59.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=
[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 60: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/60.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,
there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=
[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 61: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/61.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A.
Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=
[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 62: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/62.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=
[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 63: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/63.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=
[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 64: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/64.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so,
P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=
[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 65: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/65.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that
[~y]B = P−1~y .
Thus[T (~x)
]B
=
[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 66: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/66.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=
[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 67: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/67.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=
[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 68: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/68.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=[A~x]B
=
P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 69: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/69.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=[A~x]B
= P−1(A~x)
=
P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 70: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/70.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)=
D(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 71: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/71.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)=
D[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 72: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/72.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 73: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/73.jpg)
Connection with Diagonalization
Let A be a diagonalizable n × n matrix. Define Rn T−→ Rn by T (~x) = A~x .
Since A is diagonalizable, there is an eigenbasis assoc’d with A; that is,there is a basis B = {~v1, ~v2, . . . , ~vn} for Rn such that each vector ~vi is aneigenvector for A. Assume A~vi = λi~vi . Then
A = PDP−1 where P =[~v1 ~v2 . . . ~vn
]and D =
λ1 0 . . . 00 λ2 . . . 0...
...... 0
0 0 . . . λn
.
Recall that P = PEB; so, P−1 = PBE which means that[~y]B = P−1~y .
Thus[T (~x)
]B
=[A~x]B
= P−1(A~x)
= P−1(PDP−1~x
)= D
(P−1~x
)= D
[~x]B.
This says that D =[T]B =
[T]BB.
Section 4.7 The Matrix of an LT 27 March 2017 6 / 7
![Page 74: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/74.jpg)
A 3× 3 Example
The matrix A =
4 −1 0−1 5 −10 −1 4
has simple eigenvalues 3, 4, 6 with
associated eigenvectors
~v1 =
111
, ~v2 =
−101
, ~v3 =
1−21
.
Since B = {~v1, ~v2, ~v3} is a basis for R3, A is diagonalizable with
A = PDP−1 where P =
1 −1 11 0 −21 1 1
and D =
3 0 00 4 00 0 6
.
Here D is the B-matrix for the LT ~x 7→ A~x .
Section 4.7 The Matrix of an LT 27 March 2017 7 / 7
![Page 75: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/75.jpg)
A 3× 3 Example
The matrix A =
4 −1 0−1 5 −10 −1 4
has simple eigenvalues 3, 4, 6 with
associated eigenvectors ~v1 =
111
, ~v2 =
−101
, ~v3 =
1−21
.
Since B = {~v1, ~v2, ~v3} is a basis for R3, A is diagonalizable with
A = PDP−1 where P =
1 −1 11 0 −21 1 1
and D =
3 0 00 4 00 0 6
.
Here D is the B-matrix for the LT ~x 7→ A~x .
Section 4.7 The Matrix of an LT 27 March 2017 7 / 7
![Page 76: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/76.jpg)
A 3× 3 Example
The matrix A =
4 −1 0−1 5 −10 −1 4
has simple eigenvalues 3, 4, 6 with
associated eigenvectors ~v1 =
111
, ~v2 =
−101
, ~v3 =
1−21
.
Since B = {~v1, ~v2, ~v3} is a basis for R3, A is diagonalizable with
A = PDP−1 where P =
1 −1 11 0 −21 1 1
and D =
3 0 00 4 00 0 6
.
Here D is the B-matrix for the LT ~x 7→ A~x .
Section 4.7 The Matrix of an LT 27 March 2017 7 / 7
![Page 77: The Matrix of a Linear Transformation](https://reader031.vdocuments.mx/reader031/viewer/2022012510/6188a170b6fca1057e0d5bcc/html5/thumbnails/77.jpg)
A 3× 3 Example
The matrix A =
4 −1 0−1 5 −10 −1 4
has simple eigenvalues 3, 4, 6 with
associated eigenvectors ~v1 =
111
, ~v2 =
−101
, ~v3 =
1−21
.
Since B = {~v1, ~v2, ~v3} is a basis for R3, A is diagonalizable with
A = PDP−1 where P =
1 −1 11 0 −21 1 1
and D =
3 0 00 4 00 0 6
.Here D is the B-matrix for the LT ~x 7→ A~x .
Section 4.7 The Matrix of an LT 27 March 2017 7 / 7