Math 317: Linear Algebra
Exam 2
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
3.4. Good luck!
1. (a) (10 points)
Let A and B be 3 × 3 matrices such that the first column of B is
 
3
given by 2, and the first row of A is given by 1 0 −1 . Prove that A and
1
B cannot be inverses of each other. That is B 6= A−1 .
(b) (10 points) Suppose that A is an n × n matrix satisfying A2 − 2A + I = 0,
where I denotes the identity matrix. Prove that A is invertible.
1
Math 317: Linear Algebra
Exam 2
Spring 2016
2. Consider the following matrix:


2
1 3
A =  4 −1 3 .
−2 5 5
(10 points) Find the LU factorization of A.
2
Math 317: Linear Algebra
Exam 2
Spring 2016
3. (25 points) Consider the following matrix A and the echelon form of the augmented
matrix [A|b].

1
0
A=
1
2
1
1
1
1

2 0
0
1 −1 −1
,
2 1
2
3 −1 −3

1
 0
[U |b̂] = 
 0
0
1
1
0
0

2 0
0
b1

b2
1 −1 −1


0 1
2
−b1 + b3
0 0
0 −4b1 + b2 + 2b3 + b4
(a) Find a basis and the dimension of C(A).
(b) Find a basis and the dimension of R(A).
(c) Find a basis and the dimension of N (A).
(d) Find a basis and the dimension of N (AT ).
(e) Find a matrix X so that N (X) = C(A).
3
Math 317: Linear Algebra
Exam 2
Spring 2016
4. (a) (5 points) State the definition of V ⊥ .
(b) (5 points) Suppose that A is a symmetric n × n matrix. Let V ⊂ Rn be a
subspace with the property that if x ∈ V then Ax ∈ V . Suppose that y ∈ V ⊥ .
Prove that Ay ∈ V ⊥ .
4
Math 317: Linear Algebra
Exam 2
Spring 2016
5. (a) (10 points) Let A be an m × n matrix, v1 , v2 , . . . , vk ∈ Rn and suppose that
dim(C(A)) = n. Prove that if {v1 , v2 , . . . , vk } is linearly independent, then
{Av1 , Av2 , . . . , Avk } is linearly independent.
(b) (10 points) Let x1 , x2 , . . . , xk be vectors in Rn . Suppose there is a matrix Am×n
for which
{Ax1 , Ax2 , . . . , Axk } is linearly independent. Prove that {x1 , x2 , . . . , xk } is
linearly independent.
5
Math 317: Linear Algebra
Exam 2
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) If A is a skew-symmetric matrix, then the diagonal entries of A are 0.
(b) Let V = {x = (x1 , x2 , x3 )|x3 ≥ 0} ⊂ R3 . Then V is a subspace in R3 .
 
−1

(c) There is a 3 × 3 matrix, A, such that [1, −1, 1] ∈ R(A) and 0  ∈ N (A).
1
(d) Suppose
 that
  A is a3× 3 matrix such that a basis for N (A) is given by
0
0 
 1





0 , 1 , 0 . Then A is the zero matrix.
B=


0
0
1
6