MTH501 Official Handouts

TABLE OF CONTENTS :

Lesson 1: Introduction and Overview .................................................................................... 2 Lesson 2: Introduction to Matrices ...................................................................................... 10 Lesson 3: Systems of Linear Equations ............................................................................... 23 Lesson 4: Row Reduction and Echelon Forms .................................................................... 36 Lesson 5: Vector Equations ................................................................................................. 46 Lesson 6: Matrix Equations ................................................................................................. 62 Lesson 7: Solution Sets of Linear Systems .......................................................................... 74 Lesson 8: Linear Independence ........................................................................................... 84 Lesson 9: Linear Transformations ....................................................................................... 94 Lesson 10:The Matrix of a Linear Transformation ............................................................ 106 Lesson 11: Matrix Operations ............................................................................................ 116 Lesson 12: The Inverse of a Matrix ................................................................................... 129 Lesson 13: Characterizations of Invertible Matrices ......................................................... 143 Lesson 14: Partitioned Matrices......................................................................................... 157 Lesson 15: Matrix Factorizations....................................................................................... 175 Lesson 16: Iterative Solutions of Linear Systems .............................................................. 192 Lesson 17: Introduction to Determinant ............................................................................ 203 Lesson 18: Properties of Determinants .............................................................................. 219 Lesson 19: Cramer’s Rule, Volume, and Linear Transformations .................................... 234 Lesson 20: Vector Spaces and Subspaces .......................................................................... 245 Lesson 21: Null Spaces, Column Spaces, and Linear Transformations ............................ 269 Lesson 22: Linearly Independent Sets; Bases .................................................................... 292 Lesson 23: Coordinate System........................................................................................... 308 Lesson 24: Dimension of a Vector Space .......................................................................... 323 Lesson 25: Rank ................................................................................................................. 341 Lesson 26: Change of Basis ............................................................................................... 359 Lesson 27: Applications to Difference Equations ............................................................. 366 Lesson 28: Eigenvalues and Eigenvectors ......................................................................... 374 Lesson 29: The Characteristic Equation ............................................................................ 384 Lesson 30: Diagonalization................................................................................................ 396 Lesson 31: Eigenvectors and Linear Transformations ....................................................... 406 Lesson 32: Complex Eigenvalues ...................................................................................... 413 Lesson 33: Discrete Dynamical Systems ........................................................................... 419 Lesson 34: Applications to Differential Equations ............................................................ 429 Lesson 35: Iterative Estimates for Eigenvalues ................................................................. 442 Lesson 36: Revision ........................................................................................................... 451 Lesson 37: Revision ........................................................................................................... 452 Lesson 38: Inner Product ................................................................................................... 453 Lesson 39: Orthogonal and Orthonormal set ..................................................................... 460 Lesson 40: Orthogonal Decomposition ............................................................................. 477 Lesson 41: Orthogonal basis, Gram-Schmidt Process, Orthonormal basis ....................... 485 Lesson 42: Least Square Solution ...................................................................................... 493 Lesson 43: Inner Product Space ......................................................................................... 502 Lesson 44: Application of inner product spaces ................................................................ 512 Lesson 45: Revision ........................................................................................................... 523

Lecture 1

Introduction and Overview

What is Algebra?

History

Algebra is named in honor of Mohammed Ibn-eMusa al-Khowârizmî. Around 825, he

wrote a book entitled Hisb al-jabr u'l muqubalah, ("the science of reduction and

cancellation"). His book, Al-jabr, presented rules for solving equations.

Algebra is a branch of Mathematics that uses mathematical statements to describe

relationships between things that vary over time. These variables include things like the

relationship between supply of an object and its price. When we use a mathematical

statement to describe a relationship, we often use letters to represent the quantity that

varies, since it is not a fixed amount. These letters and symbols are referred to as

variables.

Algebra is a part of mathematics in which unknown quantities are found with the help of

relations between the unknown and known.

In algebra, letters are sometimes used in place of numbers.

The mathematical statements that describe relationships are expressed using algebraic

terms, expressions, or equations (mathematical statements containing letters or symbols

to represent numbers). Before we use algebra to find information about these kinds of

relationships, it is important to first introduce some basic terminology.

Algebraic Term

The basic unit of an algebraic expression is a term. In general, a term is either a product of a number and with one or more variables.

For example 4x is an algebraic term in which 4 is coefficient and x is said to be variable.

Study of Algebra

Today, algebra is the study of the properties of operations on numbers. Algebra

generalizes arithmetic by using symbols, usually letters, to represent numbers or

unknown quantities. Algebra is a problem-solving tool. It is like a tractor, which is a farmer's tool. Algebra is a mathematician's tool for solving problems. Algebra has

applications to every human endeavor. From art to medicine to zoology, algebra can be a

tool. People who say that they will never use algebra are people who do not know about

algebra. Learning algebra is a bit like learning to read and write. If you truly learn

algebra, you will use it. Knowledge of algebra can give you more power to solve

problems and accomplish what you want in life. Algebra is a mathematicians’ shorthand!

Algebraic Expressions

An expression is a collection of numbers, variables, and +ve sign or –ve sign, of operations that must make mathematical and logical behaviour.

For example

x x − is an algebraic expression.

What is Linear Algebra?

One of the most important problems in mathematics is that of solving systems of linear

equations. It turns out that such problems arise frequently in applications of mathematics

in the physical sciences, social sciences, and engineering. Stated in its simplest terms, the

world is not linear, but the only problems that we know how to solve are the linear ones.

What this often means is that only recasting them as linear systems can solve non-linear

problems. A comprehensive study of linear systems leads to a rich, formal structure to

analytic geometry and solutions to 2x2 and 3x3 systems of linear equations learned in

previous classes.

It is exactly what the name suggests. Simply put, it is the algebra of systems of linear

equations. While you could solve a system of, say, five linear equations involving five

unknowns, it might not take a finite amount of time. With linear algebra we develop

techniques to solve m linear equations and n unknowns, or show when no solution exists.

We can even describe situations where an infinite number of solutions exist, and describe

them geometrically.

Linear algebra is the study of linear sets of equations and their transformation properties.

Linear algebra, sometimes disguised as matrix theory, considers sets and functions, which

preserve linear structure. In practice this includes a very wide portion of mathematics!