Practice Exam II
MATH 3376.001 and 005
October 5th and 6th , 2025
Name:
Instructions:
• Answer each question to the best of your ability.
• Write all answers clearly. Partial credit cannot be awarded unless your work is clear and legible. Be
sure to cross out or erase any work you do not want graded.
Honor Code Agreement
On my honor, I have neither given nor received any aid on this exam.
Signed:
(full name)
Questions:
1
2
3
4
5
6
Total
Points:
10
10
10
15
10
10
65
Score:
Percentage
1
1. (10 points) Mark each statement as True or False.
(a) If S is a subspace of a vector space V , then S must contain the zero vector.
A. True B. False
(b) If the dimension of a vector space V is 3, then any set of 4 vectors in that space cannot span V .
A. True B. False
(c) Let A be a 3 × 5 matrix. If the column space Col(A) has dimension 3, then the null space N (A)
must be trivial, i.e., N (A) = {0}.
A. True B. False
(d) Let A be a 3 × 3 matrix. If the column vectors of A are linearly independent, then the column
vectors form a basis for R3 .
A. True B. False
1
−1
3
(e) The scalar projection of
onto
is equal to .
1
4
5
A. True B. False
2
2. (10 points) Determine whether the following set of vectors is linearly independent or dependent. Justify
your answer.
4
1
2
−6
−3
0
v1 =
−2 , v2 = 0 , v3 = −2 .
8
3
2
3
3. (10 points) Determine whether the following set of vectors spans R3 . Justify your answer.
−1
0
3
5
v1 = −1 , v2 = 2 , v3 = 4 , v4 = 6 .
1
−2
−3
−6
4
4. (15 points) Let A be the matrix
1
A = 1
0
−1
−3
2
−3
−2
−1
0
4 .
−3
(a) (10 points) Find a basis for the null space of A.
(b) (5 points) What is the dimension of the row space of A? Justify your answer.
5
5. (10 points) Let S be the subspace of R3 spanned by the vectors
1
−2
2
v1 = 2 , v2 = −5 , v3 = 5 ,
−1
−1
1
Find a basis for S ⊥ , the orthogonal complement of S in R3 .
6
1
v4 = 1 .
−4
6. (10 points) Recall the 3R planar robot arm model: if the joint angles are θ1 , θ2 ,θ3 , then the end-effector
velocity satisfies
˙
θ1
ẋ
= J θ˙2
ẏ
θ˙3
where J is the Jacobian matrix relating joint velocities to end-effector velocities.
(a) (10 points) Suppose the robot is at a configuration where its Jacobian matrix is
1
2 −1
J=
.
−1 −2 1
˙
θ1
Find all joint velocities θ˙2 that yield zero end-effector velocity.
θ˙3
(b) (5 points) (Extra Credit.) At the same configuration as in part (b), is it possible for the robot to
move its end-effector directly upwards (i.e., with ẋ = 0 and ẏ > 0)? Justify your answer.
7
0
