Math 123
Practice Final Exam
May 12, 2020
NAME:
DIRECTIONS:
• Do not open Exam until instructed to do so.
• Do each of the problems and show all work.
• NO WORK MEANS NO POINTS!
• Box or circle and LABEL your final solution.
• You will have 135 minutes to complete this Exam.
SCORES:
1. /20
2. /10
3. /15
4. /15
5. /20
6. /20
7. /10
8. /10
9. /10
Total: /130
1. 20 points Solve
(a) ˙x = −x
2
t
(b) 2xy0 − y = x for x > 0.
1
2. 10 points Use the properties of matrices to prove that multiplication by a m × n matrix A
is a linear transformation from R
n
to R
m.
To be more formal, let v ∈ R
n and LA(v) = Av. Explain why Av ∈ R
m and show LA(v)
satisfies the definition of a linear transformation.
2
3. 15 points Let A =

1 2 3
4 5 6
and B =

1 2
3 4
.
(a) Find AB.
(b) Find BA.
(c) Find B−1
.
3
4. 15 points Let
A =

0 0 0 −9 1
−2 0 0 0 0
0 0 −3 0 0
0 −1 0 0 0
1 0 0 0 6

.
(a) Using any method, calculate the determinant of A, det(A).
(b) Is A invertible? Why or why not?
(c) How many solutions does Ax = 0 have? Why?
(e) Extra Credit: Find the inverse of A.
4
5. 20 points Let
x¨ − 2 ˙x + x = 0.
(a) Verify x1(t) = e
t and x2(t) = tet are solutions.
(b) Prove S = {x1, x2} is a linearly independent set of solutions.
(c) What is the dimension of the solution space?
5
6. 20 points Let
A =

−11 8
−8 9
.
Find the solution to
˙

x
y

= A

x
y

with x(0) = 3 and y(0) = 4.
6
7. 10 points Let
x¨ − x = e
t
with x(0) = ˙x(0) = 0. Find the impulse response and use a convolution to solve for x(t).
7
8. 10 points
˙

x
y

=

5 −1
4 1  x
y

, P =

1 1
2 1
. Let 
u
w

= P
−1

x
y

=

−1 1
2 −1
 x
y

=

−x + y
2x − y

.
(a) Prove
˙

u
w

= P
−1
˙

x
y

=

3u + w
3w

=

3 1
0 3 u
w

.
(b) Solve the system in (a) for 
u
w

.
(c) Use the answer in (b) to find 
x
y

.
8
9. 10 points Two planes are given by
4x + 2y + 3z = 3 and − 3x + 4y + −3z = 4
Find the line, P0 + αv with P0, v ∈ R
3 and α ∈ R, where the two planes intersect.
9
1 Laplace Transforms
f(t) L{f(t)}(s) f(t) L{f(t)}(s)
1
1
s
e
−at 1
s + a
t
1
s
2
t
n n!
s
n+1
cos ω0t
s
s
2 + ω
2
0
sin ω0t
ω0
s
2 + ω
2
0
cosh ω0t
s
s
2 − ω
2
0
sinh ω0t
ω0
s
2 − ω
2
0
u(t − a)
e
−as
s
δ(t − a) e
−as
Table 1: Laplace Transforms
Theorem 1. Let y
0 + p(x)y = q(x) with p(x) and q(x) continuous. Then
y(x) = Ce−
R x
x0
p(t) dt + e

R x
x0
p(t) dt Z x
x0
e
R t
x0
p(s) ds
q(t) dt
for any real number C.
10

EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

E-mail: easydue@outlook.com  微信:easydue

EasyDue™是一个服务全球中国留学生的专业代写公司