Math 3353, Spring 2015
Due February 18
Homework 5 – Linear Independence and Linear Transformations
1.
Determine whether each of the following statements is True or False. If any item is False,
give a specific counterexample to show that the statement is not always true.
(a) If ~v1 , . . . , ~v4 are in R4 and ~v3 = 2~v1 + ~v2 , then the set {~v1 , ~v2 , ~v3 , ~v4 } is linearly dependent.
(b) If ~v1 and ~v2 are in R3 and ~v2 6= c~v1 for any constant c ∈ R, then the set {~v1 , ~v2 } is linearly
independent.
(c) If ~v1 , ~v2 and ~v3 are in R5 and ~v3 is not a linear combination of ~v1 and ~v2 , then {~v1 , ~v2 , ~v3 }
is linearly independent.
(d) If {~v1 , ~v2 , ~v3 , ~v4 } ⊂ R4 is linearly dependent, then {~v1 , ~v2 , ~v3 } is also linearly dependent.




3 2 10 −6
−1
 1 0 2 −4 


 and ~b =  2 . Is ~b in the range of the linear transformation
2. Let A = 
 0 1 2


3
−1 
1 4 10
8
4
T (~x) = A~x? Explain your reasoning.
An affine transformation T : Rn → Rm has the form T (~x) = A~x + ~b, where A ∈ Rm×n
and ~b ∈ Rm . Prove that T is a linear transformation if and only if ~b = ~0. Hint: break this into
two parts, (a) assume ~b = ~0 and prove that T is a linear transformation, (b) assume ~b 6= ~0 and
prove that T is not a linear transformation.
3.
4.
Consider the following transformation:
 


x1
x1 + 2x3
x2 
  

T
x3  = 4x1 − 2x2 + x4 .
x3 + x4
x4
(a) What is the domain of T ?
(b) What is the codomain of T ?
(a) Show that T is a linear transformation by finding the standard matrix that implements
the mapping.
5.
Let T : Rn → Rm be a linear transformation, with A its standard matrix. Complete the
following statements, for each one the answer is either m or n:
(a) T is one-to-one if and only if A has
(b) T maps Rn onto Rm if and only if A has
6.
pivot columns.
pivot columns.
MATLAB: Consider the following matrices,




4 −7
3
7
5
9 43
5
6 −1
2

 6 −8
5 12 −8 
 14 15 −7 −5

4
5 


 −7 10 −8 −9 14 
.



−8
−6
12
−5
−9
−3
,
B
=
A=


4
2 −6 

 3 −5
 −5 −6 −4
9
8
4 

 −5
6 −6 −7
3
13 14 15
3 11
0
1 −2
3 −1
0
Let T (~x) = A~x and S(~x) = B~x be the linear transformations associated with each matrix.
(a) Is T one-to-one?
(b) Does T map R5 onto R6 ?
(c) Is S one-to-one?
(d) Does S map R6 onto R5 ?