MA 260 LINEAR ALGEBRA
LECTURE 1
WING HONG TONY WONG
System of linear equations
This lecture is related to Sections 1.1 and 1.3.
Recall that an n-dimensional rectangular coordinate space is
Rn = cfw_(a1 , a2 , . . . , an ) : a1 , . . . , an R.
A point in t
MA 260 LINEAR ALGEBRA
LECTURE 6
WING HONG TONY WONG
Last lecture, we nished discussing about matrix multiplication and other operations, like
taking transpose and trace. We also go through some of the algebraic properties of matrix
operations, and dened t
MA 260 LINEAR ALGEBRA
LECTURE 5
WING HONG TONY WONG
Last lecture, we discussed about addition, subtraction, scalar multiplication and multiplication of matrices.
1.3 Matrices and matrix operations
Let
x1
x2
x=.
.
.
xc
be a column vector of length c. If
MA 260 LINEAR ALGEBRA
LECTURE 4
WING HONG TONY WONG
Last lecture, we nished discussing elementary row operations, Gauss-Jordan elimination
and reduced row echelon form.
1.3 Matrices and matrix operations
In general, a matrix is simply an array of numbers.
MA 260 LINEAR ALGEBRA
LECTURE 2
WING HONG TONY WONG
System of linear equations
This lecture is related to Sections 1.1 and 1.2.
Last lecture, we gave the general form of a linear equation in n variables and discussed
that its solution set forms an (n 1)-d
MA 260 LINEAR ALGEBRA
LECTURE 3
WING HONG TONY WONG
1.2 Reduced row echelon form and Gauss-Jordan elimination
Last lecture, we discussed how to use the three rules of elementary row operations on the
augmented matrix to nd the solution set for a system of
MA 260 LINEAR ALGEBRA
LECTURE 7
WING HONG TONY WONG
Last lecture, we nished the introduction to matrix inverses.
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
MA 260 LINEAR ALGEBRA
LECTURE 8
WING HONG TONY WONG
Last lecture, we introduced elementary matrices, and performing an elementary row operation on A is the same as multiplying the corresponding elementary matrix E in front of
A. Also recall that all eleme
MAT 260 LINEAR ALGEBRA
LECTURE 17
WING HONG TONY WONG
Last Wednesday, we dened determinants by cofactor expansion.
2.2, 2.3 Determinants by row reduction and column reduction and
properties of determinants
Remember that we can obtain the determinant of ma
MAT 260 LINEAR ALGEBRA
LECTURE 19
WING HONG TONY WONG
Last lecture, we proved how row operations will aect determinants of a matrix.
2.3 Properties of determinants
Theorem 1. (a ) Let B be a matrix obtained by switching two rows or two columns of A.
Then
MAT 260 LINEAR ALGEBRA
LECTURE 13
WING HONG TONY WONG
Last lecture, we discussed about diagonal, upper triangular, lower triangular, symmetric
and skew-symmetric matrices.
2.1 Determinants by cofactor expansion
The determinant of a 1 1 matrix is the entry
MA 260 LINEAR ALGEBRA
LECTURE 11
WING HONG TONY WONG
1.6 More on linear systems and invertible matrices
If we are solving multiple equations Ax = b1 , Ax = b2 , . . . , Ax = bk at the same time,
then we can row reduce (A|b1 |b2 | |bk ).
Example 1. Solve t
MA 260 LINEAR ALGEBRA
LECTURE 9
WING HONG TONY WONG
Last lecture, we proved that there are a number of equivalent statements for saying that
A is invertible. We also introduced the method to nd the matrix inverse of A in general.
1.5 Elementary matrices a
MA 260 LINEAR ALGEBRA
LECTURE 10
WING HONG TONY WONG
1.6 More on linear systems and invertible matrices
If we are solving multiple equations Ax = b1 , Ax = b2 , . . . , Ax = bk at the same time,
then we can row reduce (A|b1 |b2 | |bk ).
Example 1. Solve t