Linear Algebra 2009
Chapter 4
Chapter 4 Orthogonality
4.1 Orthogonality of the Four Subspaces
Interpretation of Ax = b
From Chapter 2 (first level)
Ax is multiplication of some numbers
From Chapter 2,3 (second level)
Ax is a combination of column vecto
Using Matlab to solve some problems in HW2
When we are trying to solve a problem for example: Sec 22.12
2x + 3y + z = 8
4x + 7y + 5z = 20 =>
A
x=b
2 3 1x
8
4 7 5 y = 20
0 2 2 z
0
0 x 2y + 2z = 0
There are many commands we can use to solve this problem.
W
DEE2485
Linear Algebra
January 15, 2009, 18:00~20:00 at ED220
Final Exam solution
1.[30%]
(1.)
2. [16%]
(1.)
=>
=>
(2.) AD
(3.) ABD
a)
rank(B)=2
b)
det(BTB)=0
c)

d)
eigenvalues of (B2+I) is 1, ,
(4.) ABCD
a)
T(T(v)=T(v)=v
b)
T(T(v)=T(v+(1,1)=( v+(1,
DEE2485
Linear Algebra
November 17, 2009, 10:10~12:00 at ED220
MidTerm solution
1
(20%)
(1) (4%)
C
(2) (4%)
b1 , b2 , b3 R ; b1 , b2 R , b3=0 ;
(3) (6%)
a)
3
b)
Arbitrary number
c)
1
(4)
2
b1 , b2 = b3 R
(6%)
a
1011
0
b
The space ST of vectors x with 1 1
L9inear Algebra 2009
Chapter 7A
Chapter 7 Linear Transformations
7.1 The Idea of a Linear Transformation
A transformation T assigns an output T(v) to each input vector v in V. The
transformation is linear if it meets these requirements for all v and w:
Linear Algebra (2009)
Chapter 1
Chapter 1 Introduction to Vectors
The heart of linear algebra is in two operations both with vectors. We add
vectors to get v+w. We multiply them by numbers c and d to get cv and dw.
Combining those two operations (adding c
Linear Algebra (2009)
Chapter 2
Chapter 2 Solving Linear Equations
How to solve a system of linear equations?
a11x1 + a12 x 2 + L + a1n x n = b1
a 21x1 + a22 x 2 + L + a 2 n x n = b 2
M
a m1x1 + a m 2 x 2 + L + amn x n = b m
2.1 Vectors and Linear Equatio
Linear Algebra (2009)
Chapter 2
2.5. Inverse Matrices
Suppose A is a square matrix.
The matrix A is invertible if there exists a matrix A1 such that A1A = I and
AA1= I. An invertible matrix is called a nonsingular matrix, while a matrix that is not inv
Linear Algebra (2009)
Chapter 3
Chapter 3 Vector Spaces and Subspaces
3.1 Spaces of Vectors
Definition of the ndimensional real (or complex) vector space Rn (or Cn):
The space Rn consists of all column vectors v with n components.
4
R2,
(1,1,0,1,1) R ,
Linear Algebra 2009
Chapter 4B
4.3 LeastSquares Approximation
When m > n, there are more equations than unknowns, sometimes there is no
exact solution x such that Ax=b.
By geometry
The best we can achieve is to find the projection Ax p . The smallest
Linear Algebra 2009
Chapter 5A
Chapter 5 Determinants
5.1 The Properties of Determinants
det(A) or A: matrix a real value
Is A invertible? Solve the linear systems, then
E.g. Solving a system
ax by k1 (1)
cx dy k 2 (2)
, given a 0 .
Step 1: Use a as
Linear Algebra 2009
Chapter 5B
5.3 Cramer s Rule, Inverse & Volumes
Theorem (Cramers Rule)
If Ax=b is a system of n linear equations in n un
knowns and det( A) 0 , then it has a unique solution. This solution is
x1
det( B1 )
,
det( A)
x2
det( B2 )
,
Linear Algebra 2009
Chapter6A
Chapter 6 Eigenvalues & Eigenvectors
6.1 Introduction to Eigenvalues
Background:
(1) Ax=b can be solved by the techniques mentioned in the previous chapters,
only if the system is stationary. What if the system is dynamic?
Linear Algebra 2009
Chapter6B
6.4 Symmetric Matrices
Background
(1) Consider the projection matrix P (note that P is symmetric): P=A(ATA)1AT.
For projection onto a line, the eigenvalues are 1 and 0; the eigenvectors are
on the line (where Px = x) and pe
Linear Algebra 2009
Chapter6C
6.6 Similar Matrices
Definition: Let M be any invertible matrices, then B = M1AM is similar to A.
(No change in s) Similar matrices A and M1AM have the same eigenvalues.
If x is an eigen vector of A, then M1x is an eige
Chapter 1.1
1
3
14
26. What combination c + d produces ? Express this
2
1
8
question as two equations for the coefficients c and d in the linear
combination.
Chapter 1.2
5. Find unit vectors u1 and u 2 in the directions of v = (3,1) and
w = (2,1,2) Fi
HW3(Deadline 11/10)
In Problem 35 (6) & 36 (14) You should use Matlab to
verify your answer. You have to print out or write down the
commands and results in Matlab. Related commands you
may refer to Matlab lecture notes page 20,26,28.
31
32
33
34
3
Homework for Chapter 4
There are solutions for some of homework problems in textbook p530533.
You have to write down how you get these solutions.
If you just copy the solution from textbook p530533, you will get 0 in that problem.
4.1
28. Why is each of
Homework for Chapter 5
Additional Problems
You may refer to Page.271
A is a 3 by 3 Matrix and C is its Cofactor Matrix
det A
0
0
T
AC =
0
det A
0
0
0
det A
Problem1: Prove the equation above
(state clearly how you derive det A and how you derive 0)
Proble
Linear Algebra(2009)
Chapter 3
3.4 The Complete Solution to Ax=b
We have discussed the case when A is invertible then there is a unique solution;
if A is noninvertible, what is the complete solution x for Ax = b?
Example
x1
1 3 0 2 1
0 0 1 4 x 2 6
x