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
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
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 Fouri
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) coecient
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 str
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 combin
MATH 311 - 500QT
Practice Final: Solutions
December
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UIN:
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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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ma
MATH 311 - 500
Practice Exam 1, solutions
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MATH 311 - 500QT
Practice Exam 2, Solutions
November
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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 eige
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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I certify that
Use Gaussian elimination to nd all solutions of the following system of equations
(Note: your calculations will be easier if you rst switch the order of the rst two
equations.)
2531 + 3.732 + $3 = 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 an
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
Solut
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 vari
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 p
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 = (
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 questi
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 +
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 ).
Solut
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 a
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 corresp
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
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
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
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 vec
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