Math 317: Linear Algebra
Exam 1
Fall 2015
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
2.2. Good luck!
2 1
1 2 1
1. Let A =
,B=
.
4 3
0 1 2
(a) (5 points) Is A a nonsingular matrix? Justify your claim.
(b) (5 points) Is B a nonsingular matrix? Justify your claim.
(c) (10 points) Let C and D be n × n matrices such that CD = In where In is
the n × n identity matrix. Furthermore, suppose that C is nonsingular. Prove
that D is nonsingular. (Caution! You cannot use the fact that C is invertible.
(See directions.))
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Math 317: Linear Algebra
Exam 1
Fall 2015
2. Let x and y be vectors in R3 .
(a) (10 points) Give an example to show that x · y = 0 does not necessarily imply
that x = 0 or y = 0.
(b) (10 points) Suppose that x · y = 0 for all x ∈ R3 . Prove that y = 0.
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Math 317: Linear Algebra
Exam 1
Fall 2015
3. 20 points, 5 points each.
(a) Find a parametric equation for the line 3x1 − x2 = 0. Write your answer as
span(a) for some vector a.
(b) Let T1 : R2 → R2 be the linear transformation that projects x onto the line
3x1 − x2 = 0 in R2 . Give the standard matrix that represents T1 .
2
5
2
2
(c) Let T2 : R → R be the linear transformation such that T2
=
and
1
3
0
1
T2
=
. Give the standard matrix that represents T2 .
1
−3
(d) Give the standard matrix that represents T2 ◦ T1 .
3
Math 317: Linear Algebra
Exam 1
Fall 2015
4. (a) (10 points) Let b1 , b2 , b3 , b4 be constants in R. Find a row echelon form for
the following matrix.

1
0

1
2
1
1
1
1
2
1
1
2

b1
b2 
.
b3 
b4
 
b1
 b2 

(b) (5 points) Find constraint equations (if any) that b = 
b3  must satisfy in
b4
order for the following system of equations to be consistent:
x1 + x2 + 2x3
x2 + x3
x1 + x2 + x3
2x1 + x2 + 2x3
=
=
=
=
b1
b2
b3
b4
Hint: Use your work in (a) to help with this problem.
(c) (5 points) Determine if (1, 2, 1, 0) ∈ span {(1, 0, 1, 2), (1, 1, 1, 1), (2, 1, 1, 2)}.
Hint: Use your work in (b) to help with this problem.
4
Math 317: Linear Algebra
Exam 1
Fall 2015
Extra Space for Problem 4 if needed.
5
Math 317: Linear Algebra
Exam 1
Fall 2015
5. (20 points, 5 points each) Determine if the following statements are true or false. If
the statement is true, give a brief justification (or an example if the problem asks
for one). If the statement is false, provide a counterexample or explain why the
statement is not true.
(a) Suppose that A is an m × n matrix such that Ax = b is consistent. If m < n,
then Ax = b has infinitely many solutions.
(b) Suppose that A is an m × n matrix such that Ax = 0 has only the trivial
solution. Then Ax = b has a unique solution.
(c) Suppose that A is a 2 × 2 matrix such that A 6= 02×2 . Then A2 6= 02×2 .
x1
x1 + 2x2
2
2
(d) Let T : R → R be defined by T
=
. T is a linear transx2
0
formation.
6
Math 317: Linear Algebra
Exam 1
Fall 2015
Extra space for Problem 5 if needed.
Bonus (5 points): What is the definition of linear algebra?
7