The Hong Kong University of Science and Technology
Introduction to Linear Algebra
MATH\' 2111

Fall 2012
Section 1.9 (Through Theorem 10) The Matrix of a Linear
Transformation
Identity Matrix In is an n x n matrix with 1s on the main left to
right diagonal and 0s elsewhere. The ith column of In is labeled
ei.
EXAMPLE: 9; 9L 0;
1 0 0
I3 = I: 91 92 9
The Hong Kong University of Science and Technology
Introduction to Linear Algebra
MATH\' 2111

Fall 2012
2.2 The Inverse ef a Matrix
The inverse of a real number a is denoted by al. For example,
An n x 71 matrix A is said to be invertible if there is an n x n
matrix C satisfying TMM WM
where In is the n x n identity matrix. We call C the inverse ofA .
The Hong Kong University of Science and Technology
Introduction to Linear Algebra
MATH\' 2111

Fall 2012
2.1 Matrix Operations
Matrix Notation:
Two ways to denote m x n matrix A:
In terms of the columns of A:
A=[a1a2
In terms Qf the entries Qf A:
6111 alj am
A = an ay am
am am] amn
Main diagonal entries:
Zero matrix:
0 = 0 0 0 THE REM 1
LetA, B, and
The Hong Kong University of Science and Technology
Introduction to Linear Algebra
MATH\' 2111

Fall 2012
I
Ax=b Mollaesmgwmel)
1.5 Solutions Sets of Linear Systems
Homogeneous System:
Ax =
(A is m x n and 0 is the zero vector in R)
EXAMELE;
H
0
X1 + 10X2
H
0
2x1 + ZOXZ
Corresponding matrix equation Ax = 0:
[321:]
H
CO
_
Trivial solution:
The homoge
The Hong Kong University of Science and Technology
Introduction to Linear Algebra
MATH\' 2111

Fall 2012
1.3 VECTOR EQUATIONS
Key concepts to master: linear combinations of vectors and ai
spanning set. Vectors in R (vectors with n entries):
Geometric Description of R2
Vector : M : is the point (x1,x2) in the plane.
iR2 is the set of all points in the pl
The Hong Kong University of Science and Technology
Introduction to Linear Algebra
MATH\' 2111

Fall 2012
1. lnr in Lin rTrnfrmIn
Another way to view Ax = b:
Matrix A is an object acting on x by multiplication to produce
a new vector Ax or b.
EXAMELE;
2 4
2
3 6 =
1
1 2
Suppose A is m x n. Solving Ax = b amounts to finding all Xi
in R which are transformed i
The Hong Kong University of Science and Technology
Introduction to Linear Algebra
MATH\' 2111

Fall 2012
MATH 111
Linear Algebra
Fall 2010
Instructor: CHENG Shiu Yuen, Rm 3452. email: macheng@ust.hk, Tel #: x7411
Office hour: Tu Th 1:45 pm2:45 pm at Rm 3452 or by appointment
Course Description: Concepts, techniques and languages of Linear Algebra are essent
The Hong Kong University of Science and Technology
Introduction to Linear Algebra
MATH\' 2111

Fall 2012
WWKE TheaA Matri unation Ax
Line; rcombIna onsc hm samatrix ector
It'plicatio K m
We, ohm In a m
D fI ii n
N IfA is amatrix, with columns 31,32, ,an, and ifx is in
then roduct ofA and x, denoted by Ax, is the linear
combination of the columns of A usi
The Hong Kong University of Science and Technology
Introduction to Linear Algebra
MATH\' 2111

Fall 2012
Example: For what value(s) of the following system admits
nontrivial solutions?
4x 6 y = x
x
y
= y
Solution:
Rewrite the system as
(4 ) x
x
4
1
If
6
0 1
( + 1) 0 4
2 3 + 2 0 ,
For
6y
= 0
( + 1) y = 0
( + 1) 0 1
( + 1)
0 1
( + 1)
0
2
6
0 0 ( 4)( 1) 6
The Hong Kong University of Science and Technology
Introduction to Linear Algebra
MATH\' 2111

Fall 2012
a. A is an invertible matrix.
Inga/J (N3: (55
\/
I. AT Is an invertible matrix.
U/ sin. an new (All mm
A v MWVCVML
a. A is an invertible matrix.
Mm, acw. .a AmvaAvmm
a. A is an invertible matrix.
D. A is row equivalent to I".
U, TFIUIAA
c. A has n p