Math 317: Linear Algebra
Exam 2
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
3.4. Good luck!
1. (a) (10 points)
Let A and B be 3 × 3 matrices such that the first column of A is
 
3
given by 2, and the first row of B 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.
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Math 317: Linear Algebra
Exam 2
Fall 2015
2. Consider the following matrix:


2
1 3
A =  4 −1 3 .
−2 5 5
(a) (10 points) Find the LU factorization of A.
     
1
3 
 2





4 , −1 , 3 is a basis for R3 .
(b) (5 points) Prove that V =


−2
5
5
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Math 317: Linear Algebra
Exam 2
Fall 2015
3. (20 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 ).
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Math 317: Linear Algebra
Exam 2
Fall 2015
4. Suppose that V and W are subspaces of Rn .
(a) (5 points) State the definition of W ⊥ .
(b) (5 points) Prove that W ⊥ is a subspace of Rn .
(c) (5 points) Suppose that V ⊂ W . Show that W ⊥ ⊂ V ⊥ .
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Math 317: Linear Algebra
Exam 2
Fall 2015
5. (a) (10 points) Let A be an m × n matrix, v1 , v2 , . . . , vk ∈ Rn and suppose that
rank(A) = n. Prove that if {v1 , v2 , . . . , vk } is linearly independent, then
{Av1 , Av2 , . . . , Avk } is linearly independent.
(b) (10 points) Let A be an n × n matrix, v1 , v2 , . . . , vk ∈ Rn and suppose that
N (A) = {0}. Furthermore, suppose that {v1 , v2 , . . . , vk } is a linearly independent spanning set for Rn . Prove that {Av1 , Av2 , . . . , Avk } is a basis for Rn .
Hint: It may be helpful to recall that An×n is nonsingular if and only if for
every b ∈ Rn there is an x ∈ Rn such that Ax = b.
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Math 317: Linear Algebra
Exam 2
Fall 2015
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) If A is an m × n matrix (m 6= n) and rank(A) = n, then R(A) = Rn , where
R(A) denotes the row space of A.
(c) There exist a matrix
  A such that the row space contains (1, 1, 1) and the null
−1
space contains −1.
−1
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Math 317: Linear Algebra
Exam 2
Fall 2015
7. (5 points) (Bonus): What is the definition of linear algebra?
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