MATH 311 - 500QT
Notes
Chapter 5, Orthogonality
Solutions of some home work from Section 5.1
Problem 1:
Solution:
a) v = 14, w = 126
12 + 3 + 27
cos =
=1
12 126
So = 1
b) vT w = 0, so = /2.
Problem 2.c
Find the scalar and vector projections of v = (4, 1

MATH 311 - 500QT
Notes
Spiegel, Chapter 1, Boundary value problems
Part 1c: Separation of variables
Idea (for functions of two variables): Try u(x, y) = X(x)Y (y).
This leads to two ODEs, one for X and one for Y .
There are generally many solutions to eac

MATH 311 - 500QT
Notes
Spiegel, Chapter 2, Fourier Series and Applications
Part 2a: Denitions and Examples
Motivation: Fourier solved the heat equation using separation of variables and
was confronted by the problem of satisfying the IC
u(x, 0) =
ck cos
k

MATH 311 - 500QT
Notes
Spiegel, Chapter 2, Fourier Series and Applications
Part 2c: Other types of Fourier Series
This section will cover cosine series and sine series.
We will not cover complex Fourier Series.
Cosine and Sine series: they are needed to s

MATH 311 - 500QT
Notes
Spiegel, Chapter 2, Fourier Series and Applications
Part 2b: Convergence
The Fourier series of f is
a0 (
nx )
nx
F (x) =
+
+ bn sin
an cos
2
L
L
n=1
where the (Fourier) coecients are
an
bn
1 L
nx
=
f (x) cos
dx
L L
L
1 L
nx
=
f (x)

MATH 311 - 500QT
Notes
Spiegel, Chapter 2, Fourier Series and Applications
Part 2d: Solution of PDEs with Fourier Series
This section will cover the solution of the heat equation and the vibrating string
using Fourier Series.
Find the solution of the heat

MATH 311 Final Exam
Date: 14th December
Time: 8-10 AM
Location: LH 238
Topics
Chapter 1
1.1
1.2 - Row Echelon Form, Guassian Elimination for solving linear systems, no Gauss-Jordan
1.3 - Linear combination, Consistent equations
1.4 Matrix Product, Inverse

MATH 311 - 500QT
Practice Final: Solutions
December
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MATH 311 - 500
Exam 2, Solutions
November 3
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MATH 311 - 500
Exam 1
October 6
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MATH 311 - 500
Practice Exam 1, solutions
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MATH 311 - 500QT
Practice Exam 2, Solutions
November
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materials that were not expressly al

Math 311.503/505 - Practice for Final Exam
1. Answer TRUE or FALSE.
i) Let A be a diagonalizable n n matrix. Then AT is also diagonalizable.
ii) Let A be a singular n n matrix. Then one of the eigenvalues of A must
be equal to 0.
iii) The exponential

MATH 311 500
Exam 2
November 3
w I I
ll ilJtKJ-Il
' l 3W5: trim-a {I UIN: t'jQ ] .3. '1 2'4"? 5
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MATH 311 - 500QT
Notes
Spiegel, Chapter 1, Boundary value problems
Part 1b: Solutions of PDEs in special cases
Case A) The solution can be found by two integrations each of one variable.
Solve
x
Solution:
Write the PDE as
2z
z
+
=0
xy y
z
cfw_x
+ z = 0
y

MATH 311 - 500QT
Notes
Spiegel, Chapter 1
Boundary value problems
Part 1a: Denitions and examples
Denition
A partial dierential equation (PDE) is an equation involving a function of more
than one variable and its (partial) derivatives.
Denition
The order

MATH 311 - 500QT
Notes
Chapter 5, Scalar product
Section 5.6, Gram-Schmidt orthogonalization
The goal of this section:
Given a basis x1 , . . . , xn of a space or subspace of an inner product space, nd
an orthonormal basis u1 , . . . , un for that space.

MATH 311 - 500QT
Notes
Chapter 3, Vector Spaces
Solutions of some home work from Section 3.2
Problem 1c:
Is S = cfw_(x1 , x2 )T | x1 = 3x2 a subspace of R2 ?
Answer: yes
Proof:
To show: the closure properties hold.
We use the second version:
Let (x1 , x2

MATH 311 - 500QT
Notes
Chapter 3, Vector Spaces
Solutions of some home work from Section 3.5
Problem 1:
Find the transition matrix from cfw_u1 , u2 to cfw_e1 , e2 for each of the following
a) u1 = (1, 1)T , u2 = (1, 1)T
Solution:
(
)
1 1
U=
1
1
b) u1 =

MATH 311 - 500QT
Notes
Chapter 5, Orthogonality
Solutions of Section 5.2, Problem 1d
Problem 1d:
Determine a basis for R(AT ), N (A), R(A) and N (AT ) of the following matrix A.
1d)
There was a question about this problem concerning a basis for R(A).
Ther

MATH 311 - 500QT
Notes
Chapter 3, Vector Spaces
Solutions of some home work from Section 3.1
Problem 3:
Let C be the set of complex numbers. Dene addition on C by
(a + bi) + (c + di) = (a + c) + (b + d)i
and scalar multiplication by
(a + bi) = a + bi
Show

MATH 311 - 500QT
Notes
Chapter 5, Orthogonality
Solutions of some home work from Section 5.2
Problem 1:
For each of the following matrices, determine a basis for R(AT ), N (A), R(A) and
N (AT ).
Solution (all solutions use Gaussian elimination):
a)
(
)
3

MATH 311 - 500QT
Notes
Chapter 3, Vector Spaces
Solutions of some home work from Section 3.6
Problem 1:
For each of the following matrices, nd a basis for the row space, a basis for the
column space and a basis for the null space.
a)
1 3 2
A= 2 1 4
4 7 8

MATH 311 - 500QT
Notes
Chapter 5, Orthogonality
Solutions of some home work from Section 5.3
Problem 1b:
Find the least squares solution of the system
x1 + x2 = 10
2x1 + x2 = 5
x1 2x2 = 20
The corresponding matrix is
1
1
1
A= 2
1 2
10
b= 5
20
The RHS is

MATH 311 - 500QT
Notes
Chapter 5, Orthogonality
Solutions of some home work from Section 5.5
Problem 1:
In b) and c), the vectors are not orthogonal to each other.
In a) and d), show that the vectors are orthogonal to each other and that they
are unit vec

MATH 311 - 500QT
Notes
Chapter 5, Orthogonality
Solutions of some home work from Section 5.4
Problem 5:
Show that for A, B Rmn the expression
A, B =
m
n
ai,j bi,j
i=1 j=1
is an inner product.
Solution
I.
A, A =
m
n
ai,j ai,j =
i=1 j=1
m
n
a2 0
i,j
i=1 j=1

MATH 311 - 500QT
Notes
Chapter 6, Eigenvalues and eigenvectors
Solutions of some home work from Section 6.1
Problem 6:
Let be an eigenvalue of A and x be an eigenvector belonging to .
Show that, for m 1, m is an eigenvalue of Am with eigenvector x.
Soluti

MATH 311 - 500QT
Notes
Chapter 6, Eigenvalues and eigenvectors
Section 6.1, Eigenvalues and eigenvectors
Denition
Let A be a square matrix.
1) A scalar is an eigenvector of A if there is a nonzero vector x such that
Ax = x.
2) A nonzero vector x is an eig

MATH 311 - 500QT
Notes
Chapter 3, Vector Spaces
Section 3.6, Row and Column spaces
Denition
Let A be an m n-matrix.
1) The subspace of Rn spanned by the rows of A is called the row space of A
2) The subspace of Rm spanned by the columns of A is called the