Name:
CSU ID:
Homework 7
October 16, 2015
1. Let LA (~x) = A~x and LB (~x) = B~x be define by matrix multiplication
with A and B defined below.
(a) Find the standard matrices for LA ◦ LB and LB ◦ LA and evaluate
LA ◦ LB (~x) where ~x = [3, 5, −7]T .
(b) Is the mapping LA ◦ LB onto?
(c) If we were interpreting the mapping LA ◦ LB as a change of basis,
what basis would we be assuming ~x is in?
(d) If we were considering the operator L−1
A ◦ LB as a change of basis,
what basis are we starting in and what basis are we finishing in?


6
3 −1


0
1 ,
A= 2
4 −3
6


4
0 4


5 2 .
B =  −1
2 −3 8
2. Given A below, find the dimensions and bases for the four fundamental
subspaces of A. The basis for the column space should be as “clean”
as possible.




A=



1
1
2
0
1
4
1
−5 −5 −13
1 −3 −19 −5 


3
3 −6
4 10
18
6 

2
2
7 −1 −4
15 11 
0
0
9 −3 −8
1
3
3. Show that S is a basis for <3 .

 
 


−4
−7 
 5


 
 

S =  3 , 5 , 6 


 6
−2
−6 
4. For what values of c is S a basis for <3 ?

 
 


−4
−7 
 c


 
 

S =  3 , 5 , 6 


 6
−2
−6 
5. Determine whether or not the statement is true or false, and justify
your answer.
(a) The span of ~v1 , · · · , ~vn is the column space of the matrix whose
column vectors are ~v1 , · · · , ~vn .
(b) If R is the reduced row echelon form of A, then those column
vectors of R that contain the leading 1’s form a basis for the
column space of A.
(c) If A and B are n × n matrices that have the same row space, then
A and B have the same column space.
(d) If E is an m × m elementary matrix and A is an m × n matrix,
then the row space of EA is the same as the row space of A.
(e) If E is an m × m elementary matrix and A is an m × n matrix,
then the column space of EA is the same as the column space of
A.
(f) The system A~x = ~b is inconsistent if and only if ~b is not in the
column space of A.
6. (a) Given A below, verify that rank(A) = rank(AT )
(b) Find a basis for the row and column spaces of A. They should
be as clean as possible.
(c) Find a basis for the null space of A and AT .
(d) How are the dimensions of the four subspaces of A related to the
row and column dimension?
(e) Which of the four fundamental subspaces does ~b live in?




A=

3 −2
1 1
1
−6
4 −5 2 −1 

,
15 −10 −4 18
6 
−6
4 −11 11
0


~b = 


4
11
72
47





7. Use matrix multiplication to find the compression of ~x = [a, b]T in the
x-direction with factor k = 1/α, where α > 1. If α < 1 what would
you call this?
8. Find the standard matrix for the stated composition in <3
(a) The rotation of 30◦ about the x-axis, followed by a rotation of 30◦
about the z-axis, followed by a contraction with factor k = 1/4.
(b) A reflection about the xy-plane followed by a reflection about the
xz-plane, followed by an orthogonal projection onto the yz-plane.
9. Let B = {M1 , M2 , M3 , M4 } be the ordered basis given below. Find the
coorinates of the vector M = I2×2 relative to the ordered basis B.
"
M1 =
1 0
2 0
#
"
, M2 =
−1 5
0 2
#
"
, M3 =
4 6
8 3
#
"
, M4 =
3 −4
6
3
#
10. Let B = 1 + 2x2 , −1 + 5x + 2x3 , 4 + 6x + 8x2 + 3x3 , 3 − 4x + 6x2 + 3x3 .
Show that B is a basis for P 3 , polynomials of degree less than or equal
to 3. For the polynomial p(x) = 1 + x + x2 + x3 , determine [p]B .