Math 18 : Winter 2024
Prof. Brendon Rhoades and Prof. John Treuer
Midterm 2
Version A
Friday, March 1, 2024
Name:
Student ID:
Discussion Section:
Instructions: Please read carefully.
• This exam is closed book, but one 8.5” × 11” sheet of notes is allowed.
• You will have 50 minutes to complete the exam.
• No electronic devices (computers, calculators, . . . ) are allowed on this exam.
• No talking or collaboration is allowed during the exam. Any student caught
talking will receive a score of 0.
• Clearly indicate your answers and show all work.
• Write your name and student ID number on the exam.
• This exam has 100 points.
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2
Problem 1: [7 + 7 + 6] Let A be the following matrix:
1 1 1 1 1
1 0 −1 −2 −3
3
4 .
A = 1 2 3 4 5 which is row-equivalent to 0 1 2
0 1 2 3 4
0 0 0
0
0
a. Find a basis for Nul(A).
b. Find a basis for Col(A).
c. Find a basis for Col(AT ).
3
Problem 2: [4 + 4 + 4 + 4 + 4] Indicate whether the following statements are True
or False. No justification is needed.
a. For all 5 × 5 matrices A and B such that both A and B are invertible, the product
AB is invertible.
T#
F#
b. If A is a 7 × 7 matrix and 3 is an eigenvalue of A, the matrix A − 3I is invertible.
T#
F#
1
1
c. There exists a basis B of R such that [x]B =
, where x =
.
0
1
2
T#
F#
d. If A and B are any 4 × 5 matrices, we have rank(A + B) = rank(A) + rank(B).
T#
F#
e. We have nullity(AB) ≥ nullity(B) for any 3 × 5 matrix A and any 5 × 6 matrix B.
T#
F#
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Problem 3: [10 + 10] Consider the vector space V = M2×2 (R) of 2 × 2 real matrices
under matrix addition and scalar multiplication. For each of the following sets W ,
determine whether or not W is a subspace of V . If W is not a subspace, explain why
not. If W is a subspace, give a basis for W .
a. W = {A ∈ M2×2 (R) : det(A) = 0}.
a b
b. W =
∈ M2×2 (R) : a + b = 0 .
c d
5
3 2 3
Problem 4: [7 + 7 + 6] Let A = 0 2 1.
0 0 2
a. Find the volume of the parallelepiped in R3 determined by the origin and the three
columns of A.
−1 2 3
b. Calculate det(2A2 B T ) where B = 0 21 2 .
0 0 31
c. What is the dimension of Nul(A − 3I)?
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Problem 5: [6+7+7] Let P be the vector space of all polynomials p(t) in the variable
t with real coefficients and consider the function
T :P→P
defined by T (p(t)) = (1 + t) · p(t).
a. Is T linear? Justify your answer.
b. What is the kernel (or the nullspace) of T ? Justify your answer.
c. Does the image (or the range) of T equal P? Justify your answer.
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This page is for scratch work. If you would like this page to be graded, clearly indicate
which problem it is to be graded for.
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