Practice Problems for Exam 2
1. Given the matrix A below, using only rational numbers, find the following: (a)The dimension of each fundamental subspace. (b)A “clean”’
basis for each subspace. (c)Which subspaces are orthogonal to each
other? Show.




A=


3
0
5 −2
6
−9
1 −17
10 −18 


15 14 −3
49
25 
6 −3
16 −13
7

~b = 


−29
148
725
−225





2. Consider the bases B = {(4, 5, −1), (−1, 0, 1), (1, 7, −1)} and B 0 =
{(3, 0, −4), (−1, 2, 5), (0, −5, 1)}.
(a) Find the matrix representation for PBB0 . The representation should
be as the product of two vectors with inverses noted with a power
of -1. Do Not multiply out.
0
(b) Find the matrix representation for PBB . The representation should
be as the product of two vectors with inverses noted with a power
of -1. Do Not multiply out.
3. 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.
4. Which of the following sets of polynomials are linearly independent in
P 3?
(a) {5 − 2x + x2 + 4x3 , 2 + 3x − 6x2 + 7x3 , −1 + 5x2 + 2x3 , 11 − 9x +
a9x2 + 9x3 }
(b) {5 − 2x + x2 + 4x3 , 2 + 3x − 6x2 + 7x3 , −1 + 5x2 + 2x3 , 11 − 9x +
a9x2 + 9x3 }
5. Referring to the previous problem, which sets of polynomials form a
bases for P 3 , polynomials of degree less than or equal to 3.
6. (a) Let PB→B 0 be the operator mapping from the basis B to B 0 . Define the operator in terms of matrix multplication. Just indicate
an inverse, do not calculate it.
(b) Let T be a linear operator that may be defined as T (~x) = A3×3 ~x3×1
where A is define below. Before the mapping T is applied, a vector [~v ]B = [−2, 3, 1]TB is given. Find ~v in the standard basis and
apply T .
(c) Convert the result in part(b) to the basis B 0 .
7. True or False
(a) The set 3, 2 + x2 , 5x − 1, x2 is a basis for P2 .
(b) Let V be the vector space of 4×4 skew-symmetric matrices (AT =
−A). The dim(V ) = 10.
(c) Let V be the vector space of 4 × 4 skew-symmetric matrices. The
dim(V ) = 6.
"
(d) For T : <2 → <2 defined by T
x
y
#!
x
y
#!
"
x
, the set
4x
"
y
, the set
4x
=
#
{T (~e1 ), T (~e2 )} is a basis for <2 .
"
(e) For T :
<2
→
<2
defined by T
=
#
{T (~e1 ), T (~e2 )} is a basis for <2 .
8. Assuming that An×n has n distinct eigenvalues, show that for the
characteristic polynomial pA (t), that pA (A) = 0n×n . Hint: Why is
there a nonsingular matrix S such that S −1 AS = Λ where Λ is a
diagonal matrix with the eigenvalues of A on the main diagonal.