lecture 3 section 1-7, 1-8 and 1-9

1

1.7

© 2012 Pearson Education, Inc.

Linear Equationsin Linear Algebra

LINEAR INDEPENDENCE

7- 2 © 2012 Pearson Education, Inc.

LINEAR INDEPENDENCE - Definition

Definition: set of vectors {v1, …, vp} in Rn is said to be linearly independent if the vector equation

x1v1 + x1v1 + …+ x1v1 = 0

has only the trivial solution. Otherwise: linearly dependent

Nontrivial solution exists linear dependence relation


LINEAR INDEPENDENCE - Example

Example 1: Let , , and .

a. Is the set {v1, v2, v3} linearly independent?

b. If possible, find a linear dependence relation among v1, v2, and v3.

1

1

v 2

3

=

2

4

v 5

6

=

3

2

v 1

0

=


LINEAR INDEPENDENCE OF MATRIX COLUMNS matrix A = [a1 a2 … a3]

The matrix equation Ax = 0 can be written asx1v1 + x1v1 + …+ x1v1 = 0

Each linear dependence relation among the columns of A corresponds to a nontrivial solution of Ax = 0.

Thus, the columns of matrix A are linearly independent iff the equation Ax = 0 has only the trivial solution.

Matrix Example

Slide 1.7- 5 © 2012 Pearson Education, Inc.


SETS OF ONE VECTOR

A set containing only one vector – say, v – is linearly independent iff v is not the zero vector.

This is because the vector equation has only the trivial solution when .

The zero vector is linearly dependent because has many nontrivial solutions.

1v 0x =v 0≠

10 0x =


SETS OF TWO VECTORS


Characterization of Linearly Dependent Sets

Theorem 7: An indexed set S = {v1, …, vp} of vectors is linearly dependent iff at least one of the vectors in S is a linear combination of the others.

In fact, if S is linearly dependent and j > 1, then some vj (with v1 ≠ 0) is a linear combination of the preceding vectors, v1, …, vj-1.


SETS OF TWO OR MORE VECTORS

Proof: If some vj in S equals a linear combination of the other vectors, then vj can be subtracted from both sides of the equation, producing a linear dependence relation with a nonzero weight on vj.

v1 = c2 v2 + c3 v3

0 = (-1)v1 + c2 v2 + c3 v3 + 0v4 + … 0vp

Thus S is linearly dependent. Conversely, suppose S is linearly dependent.

If v1 is zero, then it is a (trivial) linear combination of the other vectors in S.

( 1)−



Otherwise, , and there exist weights c1, …, cp, not all zero, such that

.

Let j be the largest subscript for which .

If , then , which is impossible because

.

1v 0≠

1 1 2 2v v ... v 0p pc c c+ + + =

0jc ≠

1j = 1 1v 0c =1v 0≠



So , and1j >

1 1 1v ... v 0v 0v ... 0v 0j j j j pc c ++ + + + + + =

1 1 1 1v v ... vj j j jc c c − −= − − −

111 1v v ... v .j

j j

j j

cc

c c−

−

= − + + − ÷ ÷


SETS OF TWO OR MORE VECTORS Theorem 7 does not say that every vector in a linearly

dependent set is a linear combination of the preceding vectors.

A vector in a linearly dependent set may fail to be a linear combination of the other vectors.

Example 2: Let and . Describe the

set spanned by u and v, and explain why a vector w is in Span {u, v} if and only if {u, v, w} is linearly dependent.

3

u 1

0

=

1

v 6

0

=

Example

Example 2: Let and .

a)Describe the set spanned by u and v b)explain why a vector w is in Span {u, v} iff

{u, v, w} is linearly dependent.


3

u 1

0

=

1

v 6

0

=


SETS OF TWO OR MORE VECTORS So w is in Span {u, v}. See the figures given below.

Example 2 generalizes to any set {u, v, w} in R3 with u and v linearly independent.

The set {u, v, w} will be linearly dependent iff w is in the plane spanned by u and v.



Theorem 8: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v1, …, vp} in Rn is linearly dependent if .

Proof: Let . Then A is , and the equation Ax = 0

corresponds to a system of n equations in p unknowns.

If , more variables than equations, so there must be a free variable many solutions

p n>1v v pA = L

n p×

p n>



Theorem 9: If a set in Rn contains the zero vector, then the set is linearly dependent.

Proof: By renumbering the vectors, we may suppose v1 = 0

1v1 + 0v2 + … + 0vp = 0 Nontrivial solution linearly dependent

1{v ,..., v }pS =

Linear Transformations


A x = b

A u = 0



Domain (R4) Co-Domain (R2)

x

T(x)

b

b

T(x)

0

Range


LINEAR TRANSFORMATIONS A transformation (or function or mapping) T from Rn

to Rm is a rule that assigns to each vector x in Rn a vector T (x) in Rm

Rn is called domain of T Rm is called the codomain of T. Notation T: Rn Rm For x in Rn, the vector T (x) in Rm is called the image of

x (under the action of T ). The set of all images T (x) is called the range of T.


MATRIX TRANSFORMATIONS

For each x in Rn, T (x) is computed as Ax, where A is an mxn matrix.

Notation for a matrix transformation by x Ax Domain of T is Rn when A has n columns Co-domain of T is Rm when A has m rows The range of T is the set of all linear combinations of

the columns of A, because each image T (x) is of the form Ax.


MATRIX TRANSFORMATIONS - ExampleExample 1: Let T: Rn Rm by T(x)=Ax

a.Find T (u), the image of u under the transformation T.

b.Find an x in R2 whose image under T is b.

c.Is there more than one x whose image under T is b?

d.Determine if c is in the range of the transformation T.

1 3

3 5

1 7

A

− = −

2u

1

= −

3

c 2

5

= −


LINEAR TRANSFORMATIONS

Definition: A transformation (or mapping) T is linear if:

i. for all u, v in the domain of T:T(u + v) = T(u) + T(v)

ii. for all scalars c and all u in the domain of T:

T(cu) = cT(u) Linear transformations preserve the operations of

vector addition and scalar multiplication.


These two properties lead to the following useful facts.If T is a linear transformation, then:

T(0) = 0T(c1v1 + c2v2 +…+ cpvp) = c1T(v1) + c2T(v2) +…+ cpT( vp)



SHEAR TRANSFORMATION

Example 2: Let . The transformation

defined by T = Ax is called a shear transformation.

1 3

0 1A

=


Contraction/Dilation

Given a scalar r, define T : Rn Rn by T(x) = rx

contraction when 0 ≤ r < 1 dilation when r > 1

How to Find Matrix of Transformation

Theorem 10: T: Rn Rm is linear transformation, then a unique matrix A such that T(x) = Ax for all x in Rn

A is mxn whose jth column is the vector T(ej) where ej is the jth column of InA = [T(e1) … T(en)]

Examine 1 point in each direction!!


Existence Question for Transformations – “Onto"

Definition: A mapping T: Rn Rm is onto Rm if each b in Rm is the image of at least 1 x in Rn

i.e., range is all of the co-domain Equivalent: does Ax = b have a solution for all

b Theorem 1-4 Pivot in every row of A Theorem 1-12: T is onto iff columns of A span Rm


Uniqueness Question for Transformations – “one-to-one”

Definition: A mapping T: Rn Rm is one-to-one if each b in Rm is the image of at most 1 x in Rn

Equivalent to Ax = b has one or no solution Pivot every column of A. Theorem 1-11: T is one-to-one iff T(x) = 0 has

only the trivial solution Theorem 1-12: T is one-to-one iff columns of A are

linearly independent


Onto/One-to-One Example


Example

T(x1, x2) = (3x1+x2, 5x1+7x2, x1+3x2)

a)Find standard matrix A of T.

b)Is T one-to-one?

c)Does T map R2 onto R3?


lecture 3 section 1-7, 1-8 and 1-9

Education

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trivial linear combination

linear dependence relation

preceding vectors

vector w

dependent setstheorem

indexed set s

independent iff v