Math 184
Spring 2021
Final Exam
Name
Answer the following questions and show all of your work clearly on these
pages. For credit, show your work clearly and completely with the appropriate
notation. Wherever appropriate, box your answers. Due Date: May 20
(No later than 8:00 a.m.)
1. Decide if the given statement is true or false. No explanation is required.
(a) The following vectors are linearly independent in R4 .
v1 = (1, 3, −1, 0), v2 = (2, 9, −1, 3), v3 = (4, 5, 6, 11), v4 = (1, −1, 2, 5), v5 = (3, −2, 6, 7)
(b) A linear system is consistent if and only if there are free variables in the row-echelon
form of the corresponding augmented matrix.
(c) If A and B are invertible n × n matrices, then so is AB.
(d) The mapping T : M2 (R) → R defined by T (A) = det(A) is a linear map.
(e) The vector b
1 2
A= 5 6
9 10
is in the span of the columns of A.
4
3
7 , b = 8
12
11
(f) S is a subspace of V .
V = Mn (R), S is the set of diagonal n × n matrices.
2. Find the general solution of the given differential equation. You should provide an
explicit solution, i.e., solve for y in terms of x.
x
dy
+ 4y = x3 − x, x > 0
dx
3. Find a basis for nullspace(A), rowspace(A), and colspace(A).
1
1 −3
2
1
A= 0
1 −1 −4
4. Use the annihilator method to determine the general solution to the following equation.
y 00 + 2y 0 + 2y = 5e6x
5. Use the Laplace transform to solve the given initial-value problem.
y 00 + y = cos(3t); y(0) = 1, y 0 (0) = 0
6. Find the general solution of the given system.
−6 2
0
x =
x
−3 1
7. (a) Determine whether the given matrix is invertible or not.
" 2 1 #
−5 3
A=
−6 5
(b) Evaluate the following (L denotes “the Laplace transform”):
3
−1 (s + 1)
i. L
s4
ii. L
Z
t
e
0
t−τ
τ dτ
8. Bonus Question (This question is only for extra credit. You don’t have to do
it.)
Verify that the Cauchy-Schwarz Inequality holds for the given pair of vectors in R4 .
u = (0, −2, 2, 1), v = (−1, −1, 1, 1)
0
