Practice Problems for Exam 2
1. 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
2. For what values of c is S a basis for <3 ?


 
 

−7 
−4

 c

 
 

S =  3 , 5 , 6 


 6
−6 
−2
3. Given









−1 
3
2







B = ~v1 =  −1  , ~v2 =  7  , ~v3 =  3 



6 
1
3



2


L(~v1 ) =  −8  ,
3




5


L(~v2 ) =  0  ,
2





0


L(~v3 ) =  −2 
6



3
2
5 







0
~1 =  7  , w
~2 =  6  , w
~ 3 =  −1 
B = w



−1
0
3 
(a) For ~v = [1, 1, 1]T , find L(~v ) in the standard basis.
(b) Using the results in (a), for ~v = [1, 1, 1]T , define L(v) where the
result is in the basis B 0 . If an inverse must be calculated just
indicate where and do not evaluate it.
4. Redo part (a) of the previous problem assuming it has been restated
in terms of polynomials.
5. The eigenvalues of


−27
180
310


213
370 
A =  −30
15 −110 −192
are λ1 = −2, λ2 = 3, and λ3 = −7.
(a) Using the RREF form of A − λI, find the eigenvectors for each
eigenvalue. Each element of the eigenvectors must be an integer.
(b) Construct the matrix P whose columns are the eigenvectors of A
such that they are in the same order as the eigenvalues.
(c) Without finding P −1 what is P −1 AP ? Explain how you came up
with the result.
6. Given the matrix above, find
(a) det(A), det(A−1 )
(b) Eigenvalues of A−1
(c) det(AT ), det(2A)
(d) Eigenvalues of AT .
(e) Eigenvalues of A − 3I. Det(A − 3I).
(f) Eigenvalues of A2 .
(g) Eigenvalues of A2 − A. Det(A2 − A).
7. 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.
8. (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





9. 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?
10. 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.
11. 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
#
12. 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 .