MAT 260 LINEAR ALGEBRA
LECTURE 28
WING HONG TONY WONG
4.2 Subspaces
Let V be a vector space. A nonempty subset W of V is a subspace of V if
(1) the addition and scalar multiplication dened in W are inherited from V ;
(2) W is a vector space.
To check whet
MAT 260 LINEAR ALGEBRA
LECTURE 22
WING HONG TONY WONG
2.2, 2.3 Determinants by row reduction and column reduction and
properties of determinants
Remember that we can obtain the determinant of matrix A by expansion along any row
or any column. So it is not
MAT 260 LINEAR ALGEBRA
LECTURE 20
WING HONG TONY WONG
2.1 Determinants by cofactor expansion
The determinant of a 1 1 matrix is the entry itself. The determinant of a 2 2 matrix
ab
ab
ab
is ad bc. We also write det
=
= ad bc.
cd
cd
cd
If A is a square mat
MAT 260 LINEAR ALGEBRA
LECTURE 19
WING HONG TONY WONG
1.7 Diagonal, triangular, and symmetric matrices
A square matrix A is a diagonal matrix if aij = 0 for all i = j , i.e. all entries outside the
main diagonal are zeroes; it is an upper triangular matri
MAT 260 LINEAR ALGEBRA
HOMEWORK 1
WING HONG TONY WONG
When you work on these homework problems, please do not simply write down the numerical answers, but also the step-by-step deduction for each problem. When I grade
your quizzes, tests and nal exam, you
MAT 260 LINEAR ALGEBRA
LECTURE 1
WING HONG TONY WONG
Review Brief introduction to set notations
Most of modern mathematics are built upon set theory. Hence, to learn modern subjects
such as linear algebra, we must rst understand the basic set language.
A
MAT 260 LINEAR ALGEBRA
HOMEWORK 5
WING HONG TONY WONG
When you work on these homework problems, please do not simply write down the numerical answers, but also the step-by-step deduction for each problem. When I grade
your quizzes, tests and nal exam, you
MAT 260 LINEAR ALGEBRA
HOMEWORK 4
WING HONG TONY WONG
When you work on these homework problems, please do not simply write down the numerical answers, but also the step-by-step deduction for each problem. When I grade
your quizzes, tests and nal exam, you
MAT 260 LINEAR ALGEBRA
HOMEWORK 2
WING HONG TONY WONG
When you work on these homework problems, please do not simply write down the numerical answers, but also the step-by-step deduction for each problem. When I grade
your quizzes, tests and nal exam, you
MAT 260 LINEAR ALGEBRA
HOMEWORK 3
WING HONG TONY WONG
When you work on these homework problems, please do not simply write down the numerical answers, but also the step-by-step deduction for each problem. When I grade
your quizzes, tests and nal exam, you
MAT 260 LINEAR ALGEBRA
LECTURE 2
WING HONG TONY WONG
1.1 System of linear equations
An n-dimensional rectangular coordinate space is
Rn = cfw_(a1 , a2 , . . . , an ) : a1 , . . . , an R.
An element in this n-space is given by an ordered n-tuple (a1 , a2 ,
MAT 260 LINEAR ALGEBRA
LECTURE 3
WING HONG TONY WONG
1.2 Gaussian Elimination
Recall from the last lecture that a system of linear equations
a11 x1 + a12 x2 + + a1n xn = b1 ,
a21 x1 + a22 x2 + + a2n xn = b2 ,
.
.
.
ar1 x1 + ar2 x2 + + arn xn = br ,
is equ
MAT 260 LINEAR ALGEBRA
LECTURE 16
WING HONG TONY WONG
1.6 More on linear systems and invertible matrices
Theorem 1. A system of linear equations has zero, one or innitely many solutions. There
are no other possibilities.
Theorem 2 (1.6.23). Let Ax = b be
MAT 260 LINEAR ALGEBRA
LECTURE 13
WING HONG TONY WONG
1.5 Elementary matrices and a method for nding A1
Recall that the three elementary row operations of a matrix A are
1. Switch row i and row j ;
2. Multiply row i by a nonzero constant k ;
3. Replace ro
MAT 260 LINEAR ALGEBRA
LECTURE 4
WING HONG TONY WONG
1.2 Gaussian elimination
Once again, the recipe for solving a system of linear equations is the following:
(1) Set up an augmented matrix.
(2) Perform a sequence of elementary row operations on the augm
MAT 260 LINEAR ALGEBRA
LECTURE 25
WING HONG TONY WONG
4.1 Real vector spaces
Before we start introducing the denition of vector space, we need to rst understand the
concept of addition and scalar multiplication. Addition is a binary operation on a nonempt