MATH 4/51021 Course Notes TOC

MATH 4/51021

Theory of Matrices

Spring 2009 Lecture Notes

Contents
Disclaimer: The lecture notes below are scans of the notes that I use for class. There is no guarantee of legibility or that the notes will make sense to anyone else. Frequently, proofs and examples actually done in class will differ somewhat from those outlined in my notes.

VECTOR SPACES

General Vector Spaces

Introduction	1
Definition	2
Basic Examples	3, 4
Basic Properties	5, 6, 7

Subspaces

Definition and Criteria	8
Subspaces - Examples	9, 10
Subspaces Associated with a Matrix	11

Linear Combinations & Spanning Sets

Definitions & Basic Theorems	12, 13
Examples	14, 15, 16

Linear Independence

Definition, Alternate Characterization	17
Examples	18, 19, 20

Basis and Dimension

Definition of Basis	20
Standard Bases for Standard Examples	21
Definition of Dimension	21
Main Results	22, 23
Bases for Row and Column Spaces	24, 25
Rank of a Matrix	25
Finding a Basis Given a Spanning Set	26
Basis for Null Space/Solution Space	27
Rank Plus Nullity Theorem (Matrices)	27
Null Space Example	28
Spaces Defined By Coordinates	29
Sums of Subspaces	30
Direct Sums of Subspaces	31, 32
Coordinate Vectors	32, 33

INNER PRODUCT SPACES

General Inner Product Spaces

Definition and Basic Properties	34
Examples	35, 36

Measurement in Inner Product Spaces

Norm of a Vector	36
Properties of the Norm	37
Distance and Angles	38

Orthogonality

Definitions & Basic Results	39
Examples of Orthogonal Complements	40
Results on Orthogonal Complements	41, 42
Orthogonal Sets	42
Orthogonal Bases & Coordinates	43
Projections	44, 45
Projections onto Subspaces	45, 46
Gram-Schmidt Orthogonalization	46, 47

LINEAR TRANSFORMATIONS

Linear Transformations

Definition & Examples	48, 49
Basic Results	49
Isomorphisms	50

Kernel, Image, and Preimages

Definition of Kernel & Image	50
Kernel & Image as Subspaces	51
Image and Preimage of a Subspace	51, 52
Example: Map Defined by Matrix Multiplication	52
Rank Plus Nullity Theorem	52, 53
Singular & Non-singular Maps	54

Spanning & Linear Independence under a Linear Map

Images of Spanning Sets	55
Preimages of Linearly Independent Sets	55
Images of Linearly Independent Sets	56
Maps Sending Bases to Bases	56
Linear Operators & Invertibility	57

Matrices and Linear Maps

Matrix of a Linear Transformation	58, 59
Examples	59, 60
Linear Maps and Matrix Operations	61
Basis Change and Coordinates	61, 62
Basis Change Matrix Examples	63, 64
Basis Change & Operators; Similar Matrices	65, 66

EIGENVALUES AND EIGENVECTORS

Characteristic Polyniomial

Definition of Characteristic Polynomial	67
Examples	68, 69
Characteristic Polynomial of Linear Operator	69
Cayley-Hamilton Theorem	70
Proof of Cayley-Hamilton Theorem	71

Eigenvalues & Eigenvectors

Definitions	72
Eigenspace Belonging to an Eigenvalue	72
Characterization of Eigenvalues/Eigenspaces	73
Examples	74, 75
Algebraic & Geometric Multiplicities	76
Eigenvectors and Similar Matrices	77

Diagonalizability

Definitions	78
Operators & Matrices	78
Diagonalizing Matrices	79
Independence of Eigenvectors	80
Conditions for Diagonalizability	81
Diagonalization Algorithm	82
Examples	83, 84

Minimal Polynomial

Definition	85
Basic Properties	86
Examples	87, 88, 89
Remarks on Block Matrices	89
Further Properties	90

CANONICAL FORMS

Invariant Subspaces

Definition & Matrices	91
Examples	92
Generalized Eigenspaces	92
Remarks on Generalized Eigenspaces	93
Examples of Generalized Eigenspaces	94, 95, 96
Characteristic & Minimal Polynomials	97

Direct Sum Decompositions

Bases	98
Invariant Direct Sums & Matrices	98, 99, 100

Primary Decomposition

Polynomial Factor Decomposition	101
Characteristic & Minimal Polynomials of Restrictions	102
Primary Decomposition Theorem	102
Notation & Corollaries	103
Kernels of Polynomial Powers	104, 105
Primary Decomposition for Split Characteristic Polynomial	105, 106
Minimal Polynomial & Diagonalizability	106

Jordan Canonical Form

Jordan Block & Jordan Form Definitions	107
Existence & Uniqueness of JCF	108
Examples (Uniqueness)	109
Main Theorem - Properties of JCF	110
Notes on Proof of Main Theorem	110, 111
Remarks on Main Theorem	111
Examples	112, 113, 114
	115, 116, 117, 118

Last updated: May 6, 2009

Donald L. White
Department of Mathematical Sciences

white@math.kent.edu