Math 323
Final Exam Sample Problems
December 2, 2015
The exam will cover material from class, the homework, and the reading. Included below are some
sample problems on the material covered since the second exam (Ch. 5). It is not meant to be an
exhaustive list, and there my be types of problems on the exam that differ from these.
1. Let L : R2 → R2 be defined by
x1
x2 − x1
L
=
.
x2
2 − x2
Is L a linear transformation? Justify your answer.
2. Let


1 1
1
2 .
A = 0 1
0 −1 −2
(a) Find the eigenvalues of A.
(b) For each eigenvalue, find a basis for the corresponding eigenspace.
(c) Find a matrix Q so that A = QDQ−1 , where D is a diagonal matrix.
(d) Use the decomposition in (c) to compute A5 .
3. Let L : R2 → R2 be the linear transformation defined by
1
3 1 x1 x2
x1
.
=
L
2 4 x2 −x1 −1
x2
Find a matrix A so that L(x) = Ax for all x ∈ R2 .
4. As usual we let S = {e1 , e2 } denote the standard basis for R2 . Suppose that B = {u1 , u2 } is
another basis for R2 , and suppose we have a transition matrix from standard coordinates to
B-coordinates
4 7
P =
.
1 2
That is, for every x ∈ R2 , we have [x]B = P [x]S .
(a) Find a matrix Q such that for every x ∈ R2 , [x]S = Q [x]B .
(b) What are u1 and u2 with respect to the standard coordinates on R2 ?
(c) Suppose that L : R2 → R2 is a linear transformation and that with respect to the basis
S we have that L is represented by the matrix A = [ 01 30 ]. What matrix represents L
with respect to B?
5. Let V be a finite dimensional vector space, and let L : V → V be an invertible linear operator.
Let λ be an eigenvalue of A.
(a) Prove that λ 6= 0.
(b) Prove that
1
λ
is an eigenvalue of L−1 .
6. Let L : R3 → R4 be given by L([x, y, z]) = [x − z, x + y, y + z, x − y − 2z]. Find a basis for
ker(L) and a basis for range(L).
7.–9. Andrilli & Hecker: 5.4: #4, #7; 5.5: #4