Solution
A process of transforming A into the row-echelon form will involve the following steps:
A
7
b
0
0
0
0
0
a
d
0
0
c
0
0
f
0
0
0
e
0
h
0
0
0
g
0
7
b
0
0
0
0
0
a
0
0
0
c
0
0
f
0
0
0
e
0
h
0
0
0
g
0
7
b
0
0
0
0
0
a
0
0
0
c
0
f
0
0
Since the resulti

4. (8 points) A certain linear homogeneous DE has solutions e(7+5i)t and
e(75i)t . Use Eulers formula to show that e7t cos 5t is also a solution. In
what way are you using the fact that the DE is linear and homogeneous?
e(7+5i)t = e7t e5ti = e7t (cos 5t +

(b) (12 points) . . . method of variation of parameters.
The form of the particular solution is
y = u1 (t)et + u2 (t)et ,
where u1 and u2 are to be determined. Differentiate this
expression once and set
u01 et + u02 et = 0.
Then differentiate again, plug

Math 315: Linear Algebra
Solutions to Assignment 2
#1
Prove:
(a) If AT A = A, then A is symmetric.
(b) If A is symmetric, then A2 is symmetric.
Solution
a. We have to show that AT = A. One has
AT =
=
=
=
(AT A)T
AT (AT )T
AT A
A.
(Theorem on the transpose

3. An electrical circuit contains a capacitor with C = 105 farad, an
inductor with L = 10 henries, a resistor with R = 100 ohms, and a
voltage source with E(t) = 15 cos(10t) volts.
(a) (6 points) Write down a differential equation for the charge Q(t)
on t

Math 315
Exam #2 Solutions in Brief
1. (12 points each) Compute the general solution of each of the following
differential equations.
(a) y 00 + 6y 0 + 9y = 0.
The roots of r2 + 6r + 9 = 0 are r1 = 3 and r2 = 3, so
the general solution is
c1 e3t + c2 te3t

5. A weight with mass 0.5 kg hangs motionless from a spring with stiffness
32 N/m. Starting at time t = 0, an external force sin(t) N is applied
in the vertical direction.
(a) (10 points) Write down an initial value problem for the position
of the weight

2. Obviously the general solution of the homogeneous differential equation y 00 y = 0 is c1 et + c2 et . Now find the general solution of the
nonhomogenous differential equation
y 00 y = 4et
by two methods:
(a) (12 points) . . . method of undetermined coe

3
Additional Example for Ch. III Sec. 1B.
Matrix check
Now we will use this Proposition III. 1B.5 to explain the check for the answers to Exercise
1.55 of Ch. 1. Following Slogan III.1.8, we write Exercise 1.55 of Ch. 1 in matrix vector form and
then labe

16
Formula for Determinants
Formula V.5.6. (ii) (Determinants of triangular matrixs)
det
d2
O
d1
.
.
dr
d1
= d1 d2 dr = det
d2
0
.
.
dr
This formula is often invalid for triangular matrices, not based on the main diagonal, but on
the other diagonal:
Ex

Math 401 - Modern Algebra
Fall 2015
Instructor: Dr. Jason McCullough
Office: 337F Science Hall
Email: [email protected]
Office Phone: (609)895-5434
Webpage: http:/canvas.rider.edu
Office Hours: MWF 10:20-11:20am, Tu 12:40-1:40pm, or by appt.
Course Me

8
Additional Exercises and Warning for Ch. IV Sec. 3A.
Exercise IV.3.11. Use the Triangle Inequality for the infinity norm of the sum of two coordinate
vectors, to prove the Triangle Inequality for this linear combination of coordinate vectors:
|Ap + Bq|

15
Set-up. Observe that there is not enough information to calculate v(x, t). But, only a bound on
v(x, t) u(x, t) is needed; so set w(x, t) = v(x, t) u(x, t). Then find equations and inequalities
for w(x, t). Use Example II.8.4 as a model. Check that L1

2
Additional Example and Exercise for Ch. III Sec. 1A.
Example III.1.12. Suppose that there are a batch of objects divided into two groups or states.
Suppose that each day half the objects in State 1 go to State 2 and half remain, also each day 0.7
of the

Additional Exercises for Ch. III Sec. 2.
.9
Exercise III.2.36. Given M3 = .1
0
the steady state vector for M3 .
.1
.9
0
.1
0 . Check that M3 is a stochactic matrix. Find
.9
Check that it is a steady state vector.
Exercise III.2.37. The question is: Which

13
Additional Exercises for Ch. II Sec. 7.
Exercise II.7.16. (A Heat Equation with a heat source) Find an infinite series solution u(x, t),
to the system of equations:
uxx = 9ut + sin 3x and u(0, t) = 7 and u(1, t) = 5.
on the domain 0 x 1, and t 0.
Set-u

Additional Examples and Exercises for Math 401
Contents.
On proving things
1
Additional Example and Exercise for Ch. III Sec. 1A.
2
Additional Example for Ch. III Sec. 1B.
3
Additional Exercises for Ch. III Sec. 2.
4
Some Quickie questions for Ch. 3 .
5
A

9
Additional Exercises for Ch. II Sec. 3.
Exercise II.3.26. (a) Prove that L(x(t) = t2 x
2tx + 2x is a linear transformation. Use the
two defining equations or the single equation of Proposition II.2.1
(b) Find many solutions to t2 x
2tx + 2x = 0; the e

5
Some Quickie questions for Ch. 3
One and two minute problems.
Ch. III Sec. 1
Can write a system of linear equations in matrix-vector form.
Can do matrix-vector multiplication.
What does multiplication by a diagonal matrix do to a vector?
What does multi

12
Exercise II.6.11. Prove that an eigenspace is always a subspace.
Writing equations in linear equation form
Example II.7.8. (A Heat Equation with a heat source) Consider a (heat) function; u(x, t),
on the domain 0 x , and t 0, which satisfies this syste

7
Additional Exercises for Ch. IV Sec. 3.
1 2
and 12 =
Exercise IV.3.11. Let A =
2 1
v, such that |v| < 12 , but |A 12 | < |A v|.
1
. State/find a specific coordinate vector,
1
Exercise IV.3.12. State an example of two specific coordinate vectors v and

1
On proving things
Query. When doing an exercise, what can we assume?
Answer NOTHING
For example. Never assume that a matrix is invertible or that a matrix is a 2 2-matrix.
One knows that a matrix is invertible, only when
1. It is given (stated in the hy

There will be two midterm exams and one final exam. Each midterm count
for 20% of your final grade. The final exam counts for 40%. The final exam
will be cumulative. The exam dates are:
Exam 1 - Monday, October 12 (during lecture)
Exam 2 - Wednesday, Nove

14
Additional Exercises for Ch. II Sec. 8.
Exercise II.8.10. Given that u(x, t) = 7e49t sin 7x is a solution to the system of equations:
uxx = ut and u(0, t) = 0 = u(, t) and u(x, 0) = 7 sin 7x,
on the domain 0 x , and t 0. Prove that it is the only solut

8. State and prove simple properties of finite groups.
9. Understand when a function or operation on an equivalence relation is
well-defined.
Classroom Decorum
Out of respect for your classmates, I ask that you arrive to class on time
everyday. If you are

11
Questions and Answers on linear equations
(a) What is a main use of the proposition on basic linear transformations?
Answer. It makes it easy to spot and verify many linear transformations.
(b) Why is it useful to know that a transformation is a linear