4 4.4 © 2012 pearson education, inc. vector spaces coordinate systems
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© 2012 Pearson Education, Inc.
4
4.4
Vector Spaces
COORDINATE SYSTEMS
Slide 4.4- 2 © 2012 Pearson Education, Inc.
THE UNIQUE REPRESENTATION THEOREM
Theorem 7: Let be a basis for vector space V. Then for each x in V, there exists a unique set of scalars c1, …, cn such that
----(1)
Proof: Since spans V, there exist scalars such that (1) holds.
Suppose x also has the representation
for scalars d1, …, dn.
1{b ,...,b }n
1 1x b ... bn nc c
1 1x b ... bn nd d
Slide 4.4- 3 © 2012 Pearson Education, Inc.
THE UNIQUE REPRESENTATION THEOREM
Then, subtracting, we have ----(2)
Since is linearly independent, the weights in (2) must all be zero. That is, for .
Definition: Suppose is a basis for V and x is in V. The coordinates of x relative to the basis (or the -coordinate of x) are the weights c1, …, cn such that .
1 1 10 x x ( )b ... ( )bn n nc d c d
j jc d 1 j n
1{b ,...,b }n
1 1x b ... bn nc c
Slide 4.4- 4 © 2012 Pearson Education, Inc.
THE UNIQUE REPRESENTATION THEOREM
If c1, …, cn are the -coordinates of x, then the vector in
[x]
is the coordinate vector of x (relative to ), or the -coordinate vector of x.
The mapping is the coordinate mapping (determined by ).
1
n
c
c
n
x x
Slide 4.4- 5 © 2012 Pearson Education, Inc.
COORDINATES IN When a basis for is fixed, the -coordinate
vector of a specified x is easily found, as in the example below.
Example 1: Let , , , and
. Find the coordinate vector [x] of x relative to .
Solution: The -coordinate c1, c2 of x satisfy
nn
1
2b
1
2
1b
1
4x
5
1 2{b ,b }
1 2
2 1 4
1 1 5c c
b1 b2 x
Slide 4.4- 6 © 2012 Pearson Education, Inc.
COORDINATES IN
or
----(3)
This equation can be solved by row operations on an augmented matrix or by using the inverse of the matrix on the left.
In any case, the solution is , . Thus and
.
n
1
2
2 1 4
1 1 5
c
c
b1 b2 x
1 3c 2 2c 1 2x 3b 2b
1
2
3x
2B
c
c
Slide 4.4- 7 © 2012 Pearson Education, Inc.
COORDINATES IN See the following figure.
The matrix in (3) changes the -coordinates of a vector x into the standard coordinates for x.
An analogous change of coordinates can be carriedout in for a basis .
Let P
n
n 1{b ,...,b }n
1 2b b bn
Slide 4.4- 8 © 2012 Pearson Education, Inc.
COORDINATES IN Then the vector equation
is equivalent to ----(4)
P is called the change-of-coordinates matrix from to the standard basis in .
Left-multiplication by P transforms the coordinate vector [x] into x.
Since the columns of P form a basis for , P is invertible (by the Invertible Matrix Theorem).
n
1 1 2 2x b b ... bn nc c c
x xP B B
n
n
Slide 4.4- 9 © 2012 Pearson Education, Inc.
COORDINATES IN
Left-multiplication by converts x into its -coordinate vector:
The correspondence , produced by , is the coordinate mapping.
Since is an invertible matrix, the coordinate mapping is a one-to-one linear transformation from onto , by the Invertible Matrix Theorem.
n
1P B
1x xP B B
x x 1P B
1P B
nn
Slide 4.4- 10 © 2012 Pearson Education, Inc.
THE COORDINATE MAPPING
Theorem 8: Let be a basis for a vector space V. Then the coordinate mapping is a one-to-one linear transformation from V onto .
Proof: Take two typical vectors in V, say,
Then, using vector operations,
1{b ,...,b }n x x
n
1 1
1 1
u b ... b
w b ... bn n
n n
c c
d d
1 1 1u v ( )b ... ( )bn n nc d c d
Slide 4.4- 11 © 2012 Pearson Education, Inc.
THE COORDINATE MAPPING
It follows that
So the coordinate mapping preserves addition.
If r is any scalar, then
1 1 1 1
u w u w
n n n n
c d c d
c d c d
B B B
1 1 1 1u ( b ... b ) ( )b ... ( )bn n n nr r c c rc rc
Slide 4.4- 12 © 2012 Pearson Education, Inc.
THE COORDINATE MAPPING
So
Thus the coordinate mapping also preserves scalar multiplication and hence is a linear transformation.
The linearity of the coordinate mapping extends to linear combinations.
If u1, …, up are in V and if c1, …, cp are scalars, then
----(5)
1 1
u u
n n
rc c
r r r
rc c
B B
1 1 1 1u ... u u ... up p p pc c c c BB B
Slide 4.4- 13 © 2012 Pearson Education, Inc.
THE COORDINATE MAPPING
In words, (5) says that the -coordinate vector of a linear combination of u1, …, up is the same linear combination of their coordinate vectors.
The coordinate mapping in Theorem 8 is an important example of an isomorphism from V onto .
In general, a one-to-one linear transformation from a vector space V onto a vector space W is called an isomorphism from V onto W.
The notation and terminology for V and W may differ, but the two spaces are indistinguishable as vector spaces.
n
Slide 4.4- 14 © 2012 Pearson Education, Inc.
THE COORDINATE MAPPING Every vector space calculation in V is accurately
reproduced in W, and vice versa. In particular, any real vector space with a basis of n
vectors is indistinguishable from .
Example 2: Let , , ,
and . Then is a basis for . Determine if x is in H, and if it is,
find the coordinate vector of x relative to .
n
1
3
v 6
2
2
1
v 0
1
3
x 12
7
1 2{v ,v }1 2Span{v ,v }H
Slide 4.4- 15 © 2012 Pearson Education, Inc.
THE COORDINATE MAPPING
Solution: If x is in H, then the following vector equation is consistent:
The scalars c1 and c2, if they exist, are the -coordinates of x.
1 2
3 1 3
6 0 12
2 1 7
c c
Slide 4.4- 16 © 2012 Pearson Education, Inc.
THE COORDINATE MAPPING
Using row operations, we obtain
.
Thus , and [x] .
3 1 3 1 0 2
6 0 12 0 1 3
2 1 7 0 0 0
1 2c 2 3c 2
3
Slide 4.4- 17 © 2012 Pearson Education, Inc.
THE COORDINATE MAPPING
The coordinate system on H determined by is shown in the following figure.
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