Partial Solutions to Homework Assignment (7)
Lecture #7
1. Discuss geometrically what does T to each vector ~ :
x
0
1
1
Sol: T (~ ) =
x
0
(a) T (~ ) =
x
1
0
x1
x2
0
1
x1
=
x2
x1
x2
0 1
Sol: T (~ ) =
x
1 0
line x2 = x1 :
(b) T (~ ) =
x
x2
; it represents 9
Lecture 4. Matrix Equations and Linearity Principle
Recall that a vector in Rm consists of m ordered real numbers, and that a m n matrix
A = [~ 1 ~ 2 : ~ n ]mn consists of n ordered vectors ~ 1 ; ~ 2 ; :; ~ n in Rm ; where
a a
a
a a
a
2
3
2
3
2
3
a11
a12
MTH 253: Elementary Matrix Algebra
Professor Chao Huang
Department of Mathematics and Statistics
Wright State University
Lecture 1: Systems of Linear Equations
Systems of two linear equations with two variables
ax + by = A
:
cx + dy = B
Recall that a pai
Lecture 8: Subspaces
De nition 8.1 A set H of Rn is called a subspace of Rn if H is closed under linear
operations. More precisely, H is a subspace if it meets the following three criteria:
1. H contains ~
0:
2. For any two vectors ~ and ~ in H; ~ + ~ is
Lecture 3. Vectors in M-Dimensional Spaces
Recall that three dimensional space, often denoted as R3 ; is the set of all vectors [x; y; z] ;
i.e.,
R3 = f[x; y; z] : x; y; and z are any real numbersg :
Furthermore, the sum, dierence, dot product of two vect
Lecture 5: Matrix Algebra
Matrix additions and Scalar Multiplications
Denition 5.1. Let A = [~ 1 ; ~ 2 ; :; ~ n ]m
a a
a
[bij ]m
n
n
= [aij ]m
n
h
i
and B = ~ 1 ; ~ 2 ; :; ~ n
b b
b
=
m n
be two matrices of the same dimension, where columns
2
3
2 3
a1i
b1
Lecture 2. Solving Linear Systems
As we discussed before, we can solve any system of linear equations by the
method of elimination, which is equivalent to applying a sequence of elementary
row reductions over its augmented matrix. Some terminologies:
Lead
Lecture 10: Cramer Rules
s
Let A be a n
system
n invertible matrix whose columns are [~ 1 ; :; ~ n ] and consider linear
a
a
A~ = ~
x b:
We know that the solution is unique and may be represented as the form
b:
~ = A 1~
x
However, calculation of inverse m
Partial Solutions to Homework (1 & 2)
Homework from Lecture 1:
1. Solve the following linear systems directly by the method of elimination:
(a)
8
< x1 3x2 + 4x3 = 4
3x1 7x2 + 9x3 = 8
:
4x1 + 6x2 x3 = 7
Equ(1)
Equ(2)
Equ(3)
Solution: (Label equations as ab
Lecture 9: Determinants
For 2 2 matrix
we have learn that the quantity
a b
A=
;
c d
det (A) = ad bc
is called determinant of A:
For 3 3 matrix A; det (A) is de ned through the following "diagonal rule"
2
3
.
a11 a12 a13 . a11 a12 a13
.
6
7
.
[A; A] = 6a21
Lecture 11: Eigenvalues and Eigenvectors
De nition 11.1. Let A be a square matrix (or linear transformation). A number is
called an eigenvalue of A if there exists a non-zero vector ~ such that
u
A~ = ~ :
u
u
(1)
In the above de nition, the vector ~ is ca
Lecture 7: Linear Transformations
De nition A linear transformation is a mapping (or function) T from Rn to Rm satisfying
(i) T (~ + ~ ) = T (~ ) + T (~ ) and (ii) T (u) = T (~ ) for any real number :
u v
u
v
~
u
Example 6.1 In 1-D, T (x) = cx (c is a con
Lecture 6. Inverse of Matrix
Recall that any linear system can be written as a matrix equation
A~ = ~
x b:
In one dimension case, i.e., A is 1 1; then
Ax = b
can be easily solved as
x=
b
1
= b = A1 b provided that A 6= 0:
A
A
In this lecture, we intend to