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 independenceIt 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 transformation2D picture can be changed

Linear Transformationif 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 transformationExample of notation;

Standard matrix of mean the matrix of column A in