10 Oct 2014
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Linear Algebra@Chapter 1

Posted in Math By khlau On October 10, 2014

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

<br>

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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  1. By khlau

    Homogeneous Linear System
    => can be in form of Ax= 0 and at least one free variable

    • it must be consistent
    • free variable appears when #non-zero rows < #variables. If the #pivot columns = #columns, there will have no free variables though there are zero rows.
    • x= {x1,x2,x3}, free variable e.g x3 can make x1, x2 in term of x3,
      then e.g. x= {ax3,bx3,x3}=x3{a,b,1}=su

    => can be written as x= su + tu => parametric vector equation

    • this form is the solution of matrix. We solve the linear system to find x1,x2,x3, but now we rewrite them into parametric vector equation

    Non-homogeneous Linear System
    => x = p+ tv => general parameter

    • p is a parameter to translate the solution set of Ax=b, if x = tv for Ax=0
    • a line pass through a(point) parallel to b(line start from 0 point) , where a is {1,2} b is {3,4}, will give the parametric equation as x= {1,2}+ t{2.4}

    Theorem 6:

    • Ax=0 is consistent, v is any point on Ax=0 //v can sub. into x then make Ax=0
    • Ax=b is consistent for some b, let p are some point on Ax=b //p can sub. into x then make Ax=b
      any vector can be written as w = p + v, then w is solution of Ax=b

    • By khlau

      if A can be written as echelon form, Ax=b is consistent.

  2. By khlau

    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.

    • if Ax=0 has free variable, then we can sure Ax=0 can have non-trivial solution (which x != 0 can be solution)
    • if Ax= 0 has no free variable, then Ax is linear independence
    • if Ax = 0 has free variable, then Ax is linear dependence

    Conclude: if 0 is the only solution to make Ax = 0 , then Ax is linear independence, else it is linear dependence

  3. By khlau

    Transformation

    This use a new idea to view Ax = b.

    • A is matrix act on x to transform x into b.
      e.g A is a matrix to transfrom x with R^4 into b with R^2
    • Transformation = function = mapping = T
    • x in R^n in domain transform into T(x) in R^m in codomain. T(x) is the Image of x. Set of T(x) is call Range.
    • Notation : transform x into Ax , x \mapsto Ax

    Tips: #columns in A equals to #row in b since each column requires one coefficient

    • By khlau

      Shear transformation
      2D picture can be changed
      \mathbb{R}^{2} \mapsto \mathbb{R}^{2}

    • By khlau

      Linear Transformation

      if T is function of linear transformation, then

      • T\left ( cp_{1} +cp_{2} +cp_{3} ...cp_{n} \right ) =T\left ( cp_{1} \right )+T\left ( cp_{2} \right )+..+T\left ( cp_{n} \right )
      • if T is no linear transformation, T(0) \neq 0

    • By khlau

      One-to-one Linear Transformation

      • T has the properties of linear transformation
      • T map \mathbb{R}^{n}\rightarrow \mathbb{R}^{m} if and only if column A span \mathbb{R}^{m}, which means m pivots column are needed to ensure Ax=b is consistent for each b in \mathbb{R}^{m}
      • T is one-to-one if and only if A are linear independent
      • A is linear independent if and only of A only has the trivial solution (Ax = 0 when x = 0 )

      • By khlau

        one-to-one for \mathbb{R}^{n}\rightarrow \mathbb{R}^{m} will be possible if A has n columns , each column has m rows and m pivot columns

  4. By khlau

    Matrix Transformation

    It is talking about the implementation of mapping
    x can be view as e.g. in \mathbb{R}^{2} , x = x_{1}\begin{bmatrix}1\\0 \end{bmatrix} + x_{2} \begin{bmatrix}0\\1\end{bmatrix}

    Type of transformation

    • Reflection
    • Contraction and Expansion
    • Shear
    • Projection

    • By khlau

      Example of notation;
      T(x)
      =T(x_{1},x_{2})
      =(3x_{1}+5x_{2},2x_{1}+7x_{2})
      =\begin{bmatrix}3 &5 \\ 2 & 7\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}

    • By khlau

      Standard matrix of T mean the matrix of column A in T(x)=Ax

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