10/7/2010 FIRST HOURLY PRACTICE IV
Math 21b, Fall 10
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n
Solutions
Math 21b, Spring 09
1. The verication that cos(nx), sin(nx), 1/ 2 form an orthonormal family is a straightforward computation, when using the identities provided. For example, cos(nx), sin(mx) =
1
1
sin(nx) sin(mx) dx = 2 cos(n m)x) cos(n + m)x
FINAL PRACTICE II, December 17, 2010
Math 21b, Fall 10
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11/4/2010 SECOND HOURLY PRACTICE I
Math 21b, Fall 10
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FINAL PRACTICE I, December 17, 2010
Math 21b, Fall 10
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10/7/2010 FIRST HOURLY PRACTICE VI
Math 21b, Fall 10
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11/4/2010 SECOND HOURLY PRACTICE III
Math 21b, Fall 10
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11/4/2010 SECOND HOURLY PRACTICE II
Math 21b, Fall 10
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Chapter 2
Linear Algebra
Introduction
The purpose of this chapter is to provide sucient background in linear
algebra for understanding the material of Chapter 3, on linear systems of
dierential equations. Results here will also be useful for the developme
10/7/2010 FIRST HOURLY PRACTICE IV
Math 21b, Fall 10
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Basic Linear algebra
c A. Baker
Andrew Baker
[08/12/2009]
Department of Mathematics, University of Glasgow.
E-mail address: [email protected]
URL: http:/www.maths.gla.ac.uk/ajb
Linear Algebra is one of the most important basic areas in Mathematics,
FINAL PRACTICE II, December 17, 2010
Math 21b, Fall 10
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11/4/2010 SECOND HOURLY PRACTICE VI
Math 21b, Fall 10
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FINAL PRACTICE IV, December 17, 2010
Math 21b, Fall 10
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FINAL PRACTICE III, December 17, 2010
Math 21b, Fall 10
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10/7/2010 FIRST HOURLY PRACTICE II
Math 21b, Fall 10
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11/4/2010 SECOND HOURLY PRACTICE III
Math 21b, Fall 10
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10/7/2010 FIRST HOURLY PRACTICE I
Math 21b, Fall 10
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10/7/2010 FIRST HOURLY
Math 21b, Fall 2010
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Problem 1) TF questions (20 points) No justications are needed.
1)
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T
F
The rank of A1 is always equal to the rank
Problem Set 18 - Diagonalization
The first problem below is a warmup; as usual, you need not turn it in.
W1. (a) Bretscher #7.5.20
Solution. For this matrix A, the trace is tr A = 0 and the determinant is det A = 1. Thus the
characteristic polynomial is f
Problem Set 16 Determinants, Introduction to Discrete Dynamical
Systems and Eigenanalysis
1
2
1. Let A =
0
3
0
1
5
0
2
0
1
2
3
0
. In this problem, youll calculate det A in two ways.
1
1
(a) Calculate det A using minors by first expanding down the first
Problem Set 8 - Coordinates
1
1
1
W1. (a) Verify that the vectors ~v1 = 1 , ~v2 = 2 , ~v3 = 3 form a basis B of R3 .
1
3
6
Solution. As in #2(a) on the worksheet
More
on Bases of Rn , Matrix Products, (~v1 , ~v2 , ~v3 ) is
3
a basis of R if the matr
Problem Set 21 - Introduction to Continuous Dynamical Systems
1. Modeling population growth with differential equations.(1)
(a) The simplest model of population growth is often credited to Thomas Malthus, who suggested that
a population grows at a rate pr
Problem Set 4 - How much data do you need to determine a linear
transformation?
1. A simple but important example of a linear transformation is the linear transformation T : Rn
Rn defined by T (~x) = ~x. This is called the identity transformation. Find t