Math 317: Linear Algebra
Exam 3
Spring 2016
Name:
*No notes or electronic devices. You must show all your work to receive full credit. When
justifying your answers, use only those techniques that we learned in class up to Section
5.1. I am thinking of the number thirty seven. Unless otherwise stated, you do not have
to simplify your answers. Good luck!
1. (20 points) Suppose that A is a 4 × 4 matrix with rows a, b, c and d such that
det A = 5. That is,
 
a
b

A=
c
d
(a) (10) Calculate det B if


3a + b + d


b

B=
 c+d 
d
(b) (5) Calculate det A−1 or explain why it does not exist.
(c) (5) Calculate det AB (if possible). If it is not possible, explain why.
1
Math 317: Linear Algebra
Exam 3
Spring 2016
2. (15 points) Let V = Span ((1, 0, 1, 0), (0, 1, 4, 2)) ⊂ R4 .
(a) (5) Find an orthogonal basis for V .
(b) (5) Find a basis for V ⊥ .
(c) (5) Find the projection of (1, 1, 1, 1) on V ⊥ . Hint: Use the orthogonal basis
you obtained in (a) to your advantage.
2
Math 317: Linear Algebra
Exam 3
Spring 2016
3. (15 points) Let A and B be n × n matrices and let Rn×n denote the space of all
n × n matrices.
(a) (5) Define the trace of a matrix A.
(b) (10) Let < A, B >= trace(AB). Show that <
A, B >
is not an inner product
1 1
on Rn×n . Hint: Consider < A, A > for A =
.
−2 1
3
Math 317: Linear Algebra
Exam 3
Spring 2016
4. (15 points) Find the QR-factorization of the following matrix:

1
1
A=
1
1

2
4

2
4
4
Math 317: Linear Algebra
Exam 3
Spring 2016
 
1
3

5. (20 points total, 5 points each) Let V ⊂ R be the plane spanned by v1 = 0 and
2
 
1

v2 = 1.
0
(a) Find a vector v3 so that V ⊥ = span(v3 ). Note that B = {v1 , v2 , v3 } is a basis
for R3 .
(b) Let T : R3 → R3 be the linear transformation which reflects x ∈ R3 over V .
Find [T ]B , the matrix of T with respect to the basis B.
(c) Use the change of basis formula to compute [T ]stand . That is, compute the
standard matrix of T . You do not need to carry out the matrix multiplication/inversion.
(d) Calculate det [T ]stand .
5
Math 317: Linear Algebra
Exam 3
Spring 2016
6. (5 points each) Determine if the following statements are true or false. If the statement is true, give a brief justification. If the statement is false, provide a counterexample or explain why the statement is not true.
(a) Suppose that the columns of A contain a basis for a subspace V ⊂ Rm . If
A = QR where Q is an orthogonal matrix and R is an upper triangular matrix
with positive diagonal entries, then the projection matrix (which projects any
x ∈ Rn onto V ) is given by PV = QQT .
(b) det(A + B) = det A + det B for any two n × n matrices A and B.
(c) There is a nonsingular matrix P such that
2 2
1 1
P =P
.
0 4
0 5
(d) Suppose that A and B are n × n matrices such that trace(A) = trace(B). Then
A is similar to B.
6
Math 317: Linear Algebra
Exam 3
Spring 2016
7. (5 points) (Bonus): I am thinking of a number between 1-100. What is it?
7