MAT 211 PRACTICE MIDTERM 1
This is a practice midterm 1 for MAT 211 L03. The format of this practice midterm will be
the same as our actual midterm: 15 True/False questions, followed by 5 longer multi-part
questions. Each True/False question is worth 2 points (2 points for a correct answer, 0 points
for an incorrect or blank answer). Each of the longer multi-part questions is worth a total
of 10 or 15 points, and the points for each part are listed in the problem. There are a total
of 100 points.
You have 80 minutes to complete this practice midterm. You are encouraged to take this
practice midterm under realistic test conditions. Solutions will be posted after our midterm
review session.
For the longer multi-part questions, you should show your work and explain your reasoning.
Correct answers with no work or explanation will not receive much partial credit. Clearly
shown work / explanations with minor errors will receive partial credit, possibly close to full
credit.
For the True/False questions, just an answer (T/F) is fine.
Problem 1: True or False. (2 points each, 30 points total.)
(1) Every invertible n-by-n matrix has rank n.
(2) If the image of an m-by-n matrix A contains the standard vector ⃗e1 in Rm , then ⃗e1
is a scalar multiple of the first column of A.
(3) Let ⃗v and w
⃗ be nonzero vectors in R3 . The set of vectors ⃗x in R3 such that ⃗v ·⃗x = w
⃗ ·⃗x
is a linear subspace of R3 .
(4) If A is the matrix of a counterclockwise rotation by 90 degrees, there is a positive
integer n such that An = I2 .
(5) If A is an 3-by-4 matrix, and the first and second columns of rref(A) are linearly
dependent, then the first and second columns of A are linearly dependent.
(6) Suppose S : R2 → R2 and T : R2 → R2 are rotations. The matrix of the composition
S ◦ T is the same as the matrix of the composition T ◦ S.
(7) Let A be a nonzero 1-by-3 matrix. For any y ∈ R, the linear system A⃗x = y has
infinitely many solutions.
1
2
MAT 211 PRACTICE MIDTERM 1
(8) There is a linear subspace of R4 whose dimension is 5.
(9) If A is a 3-by-3 matrix whose kernel has dimension 1, then the image of A has dimension 2.
(10) If ⃗v1 , ⃗v2 are linearly independent vectors in R3 , then ⃗v1 − ⃗v2 , ⃗v2 − ⃗v1 are linearly independent.
2 0
(11) Let A =
. For all nonzero vectors ⃗v in R2 , the length ||Av|| is greater than
0 2
the length ||v||.
x
(12) If P is the plane in R3 given by y : x, y in R , then P contains a linear sub
1
space of R3 .
(13) Let A be a 3-by-4 matrix and let B be a 5-by-3 matrix. The products AB and BA
are both defined.
(14) If ⃗v1 , ⃗v2 , ⃗v3 , ⃗v4 are vectors in Rn whose span has dimension 2, then the vectors ⃗v2 , ⃗v3 , ⃗v4
are linearly dependent.
(15) Let A be a 2-by-3 matrix, and let B be a 3-by-2 matrix. If AB has rank 1, then the
columns of B are linearly dependent.
Problem 2: (10 points)
7
1 2 3
(a) (5 points) Let A = 0 1 2 , and let ⃗y = 3. Find all solutions to the system of
3
0 2 1
linear equations A⃗x = ⃗y .
(b) (5 points) Write down the rank of A, the kernel of A, the image of A, and state whether
A is invertible. Briefly explain your answers.
Problem 3: (15 points). For this problem, ⃗v1 , ⃗v2 , ⃗v3 , ⃗v4 are vectors in R4 , and ⃗v1 , ⃗v2 , ⃗v3 are
linearly independent. Let B be the matrix whose columns are ⃗v1 , ⃗v2 , ⃗v3 , ⃗v4 (from left to right).
(a) (3 points) What is the size of B (number of rows, number of columns)?
(b) (4 points) What are the possible dimensions of ker(B)? Give an example for each possibility.
MAT 211 PRACTICE MIDTERM 1
3
(c) (4 points) Suppose A is a 2-by-4 matrix. What are the possible values of rank(AB)?
Give an example for each possibility.
(d) (4 points) Is there a 2-by-4 matrix A, and a 4-by-2 matrix C, such that ABC = I2 ?
Explain why or why not.
Problem 4: (15 points)
x+y
x
(a) (2 points) Is the function T : R2 → R3 given by T
= y + 1 a linear transformay
x+1
tion? If so, write down the matrix of T . If not, explain why not.
x
x+y
2
2
(b) (3 points) The function S : R → R given by S
=
is a linear transfory
x+y
mation. Describe S geometrically (using at least one of the words “rotation”, “scaling”,
“projection”, “reflection”, “shear”). Draw a picture of what S does to the standard vectors
⃗e1 , ⃗e2 .
1
1
3
3
(c) (5 points) Is there a linear transformation U : R → R such that U 3 = 0,
1
2
0
1
0
0
U 1 = 1 , and U 1 = 2? Explain why or why not.
0
0
1
1
1
0
(d) (5 points) Describe the span of the vectors 3 and 1 geometrically. Find a nonzero
2
1
vector ⃗v that is orthogonal to both of these vectors.
Problem 5: (15 points)
(a) (5 points) Let P be the projection onto the x-axis, and let R be the rotation by 90
degrees counterclockwise. What is the matrix of the composition R ◦ P ?
(b) (5 points) Draw a picture of what the linear transformation R ◦ P does to the standard
vectors ⃗e1 and ⃗e2 .
(c) (5 points) What is the image of R ◦ P ? What is the kernel of R ◦ P ? Using your answers
to these questions, write down the matrix of the composition R ◦ P ◦ R ◦ P .
Problem 6: (15 points)
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MAT 211 PRACTICE MIDTERM 1
(a) (5 points) Find all 2-by-2 matrices A =
1 1
.
2 2
a b
that commute with the matrix B =
c d
a
b
a b
(b) (5 points) Let VB be the subset of vectors ⃗v = such that A =
commutes
c d
c
d
4
with B. Show that VB a linear subspace of R .
(c) (5 points) What is the dimension of VB ? Is there a 2-by-2 matrix C so that the subspace
VC (defined analogously as in part (b)) has larger dimension than VB ?
0
