10 Oct 2014
Linear Algebra@Chapter 1
The notes begin at Solution set of linear system.
The concept to start this topic is free variable, row operation and span, 1.1 to 1.4 in the book "linear algebra and its applications 4th edition"
"span" : generally means can be solved by
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save your life by using http://www.wolframalpha.com/ to solve the matrix
format : row reduce {{row1},{row2}...}
example:
row reduce {{4,6,-7,3,-5},{-7,-8,10,-5,6},{3,5,-8,4,-6},{7,12,-9,2,-7},{5,-8,14,-6,3}}
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Homogeneous Linear System
=> can be in form of Ax= 0 and at least one free variable
then e.g. x= {ax3,bx3,x3}=x3{a,b,1}=su
=> can be written as x= su + tu => parametric vector equation
Non-homogeneous Linear System
=> x = p+ tv => general parameter
Theorem 6:
any vector can be written as w = p + v, then w is solution of Ax=b
if A can be written as echelon form, Ax=b is consistent.
Linear independence
It is talking about a set of vectors, i.e #column > 1. Linear dependence means the there exist some vectors(not all) have scalar relationship, i.e. the vectors can cancel each other by tuning their scalar(coefficient or solution x1,x2....xn ) to gain 0.
Conclude: if 0 is the only solution to make Ax = 0 , then Ax is linear independence, else it is linear dependence
Transformation
This use a new idea to view Ax = b.
e.g A is a matrix to transfrom x with R^4 into b with R^2
Tips: #columns in A equals to #row in b since each column requires one coefficient
Shear transformation

2D picture can be changed
Linear Transformation
if T is function of linear transformation, then
One-to-one Linear Transformation
one-to-one for
will be possible if A has
columns , each column has
rows and
pivot columns
Matrix Transformation

It is talking about the implementation of mapping
x can be view as e.g. in
Type of transformation
Example of notation;




Standard matrix of
mean the matrix of column A in 