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

1
1.7
© 2012 Pearson Education, Inc.
Linear Equations
in 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 = [a1a2 … a3]
 The matrix equation Ax = 0 can be written as
x1v1 + 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.
1
v 0x =
v 0≠
1
0 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
.
1
v 0≠
1 1 2 2
v v ... v 0p p
c c c+ + + =
0j
c ≠
1j = 1 1
v 0c =
1
v 0≠

 So , and1j >
1 1 1
v ... v 0v 0v ... 0v 0j j j j p
c c +
+ + + + + + =
1 1 1 1
v v ... vj j j j
c c c − −
= − − −
11
1 1
v v ... v .j
j j
j j
cc
c c
−
−
   
= − + + − ÷  ÷
   

 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
 
 =
 
  

 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>
1
v vp
A  =  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 }p
S =

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 - Example
Example 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
− 
 =
 
−  
2
u
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) = 0
T(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 1
A
 
=  
 

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 In
A = [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

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