Chapter 2
First Order Dierential Equations
Introduction
Any rst order dierential equation can be written as
F (x, y, y ) = 0
by moving all nonzero terms to the left hand side of the equation, and y must appear explicitly in
the expression F . Our study of
Linear Algebra and Matrix Theory
Part 1 - Linear Systems, Matrices and Determinants
This is a very brief outline of some basic denitions and theorems of linear
algebra. We will assume that you know elementary facts such as how to add
two matrices, how to
Here we will clean up one of the proofs from class.
Suppose that V is an inner product space and T : V V satises
v = T (v ) for all v V .
We wish to show that
v , w = T (v ), T (w) for all v, w V .
To do this, we will use the polarization identity from cl
In this note we will give a counterexample to a question in class. Here is the question:
Question: Does there exist a set V , a subset W V and two elds F1 , F2 such that
1. V is a vector space over F1 and W is a subspace over F1 and
2. V is a vector space
Lecture 8
We now want to give a theorem that characterizes one-to-one and onto linear maps in a
dierent way.
Theorem 0.1. Let T : V W be linear.
1. T is one-to-one if and only if it maps linearly independent sets in V to linearly independent sets in W .
2
Math 341, Solutions to Practice for the Final
Summer 2012
1. Complete the following denitions:
[Your denitions need to be precise. Make sure you know the denitions of the given terms, not
just theorems/techniques related to the terms.]
(a) Let V be a vect
Chapter 4
Systems of Linear Dierential
Equations
1. Systems of Dierential Equations
Introduction to Systems
Up to this point the entries in a vector or matrix have been numbers. In this section, and in the
following sections, we will be dealing with vecto
Chapter 3
Second Order Linear Dierential
Equations
3.1 Introduction; Basic Terminology
Recall that a rst order linear dierential equation is an equation which can be written in
the form
y + p(x)y = q(x)
where p and q are continuous functions on some inter
DIFFERENTIAL EQUATIONS
Chapter 1
Introduction and Basic
Terminology
Most of the phenomena studied in the sciences and engineering involve processes that change with
time. For example, it is well known that the rate of decay of a radioactive material at ti
Linear Algebra and Matrix Theory
Part 4 - Eigenvalues and Eigenvectors
1. References
(1) S. Friedberg, A. Insel and L. Spence, Linear Algebra, Prentice-Hall.
(2) M. Golubitsky and M. Dellnitz, Linear Algebra and Dierential Equations Using Matlab, Brooks-C
Linear Algebra and Matrix Theory
Part 5 - Inner Products, Hermitian and Unitary Matrices, etc.
1. References
(1) S. Friedberg, A. Insel and L. Spence, Linear Algebra, Prentice-Hall.
(2) M. Golubitsky and M. Dellnitz, Linear Algebra and Dierential Equation
Linear Algebra and Matrix Theory
Part 3 - Linear Transformations
1. References
(1) S. Friedberg, A. Insel and L. Spence, Linear Algebra, Prentice-Hall.
(2) M. Golubitsky and M. Dellnitz, Linear Algebra and Dierential Equations Using Matlab, Brooks-Cole.
(
Linear Algebra and Matrix Theory
Part 2 - Vector Spaces
1. References
(1) S. Friedberg, A. Insel and L. Spence, Linear Algebra, Prentice-Hall.
(2) M. Golubitsky and M. Dellnitz, Linear Algebra and Dierential Equations Using Matlab, Brooks-Cole.
(3) K. Hom
Here we will give an example of a norm on a vector space that is not possible to construct
using an inner product.
Let V = R2 as a vector space over R and dene a norm
(x, y ) = |x| + |y | ,
where |x| is just the absolute value of a number x. We claim that