文章目录
  1. 1. 01#The Geometry of Linear Equations
    1. 1.1. a tip and enlightenment
    2. 1.2. elimination
  2. 2. 02#Elimination with Matrices

Here are some notes which I think is wonderful in lectures.


01#The Geometry of Linear Equations

a tip and enlightenment

there are two methods to calculate the result of matrix A multiply vector x, but I suggest you think as following:

AxAx is a combination of columns of A.

For example

A equals

[2513]\begin {bmatrix} 2 & 5\\ 1 & 3 \end {bmatrix}

x equals

[12]\begin {bmatrix} 1\\ 2 \end {bmatrix}

then we can get

1[21]+2[53]=[127]1 \begin {bmatrix} 2\\ 1 \end{bmatrix} +2\begin{bmatrix} 5\\ 3 \end{bmatrix}= \begin{bmatrix} 12\\ 7 \end{bmatrix}

so, it’s just like a column picture.

just image it.

elimination

we can use elimination to find the solution if there is one. we can find the solution to a system of any size and find out there is no solution if elimination fails.

02#Elimination with Matrices

what will happen if we use a row vector times a matrix?

like

[124][345126012]\begin {bmatrix} 1 & 2 &4 \end {bmatrix} \begin {bmatrix} 3&4&5\\ 1&-2&6\\ 0&1&2 \end {bmatrix}

okey. we will try to solve it.

it gives

1[345]+2[126]+4[012]=[5425]1\begin {bmatrix} 3&4&5 \end {bmatrix} +2\begin {bmatrix} 1&-2&6 \end {bmatrix} +4\begin {bmatrix} 0&1&2 \end {bmatrix} =\begin {bmatrix} 5&4&25 \end {bmatrix}

the result is still a row vector, isn’t it?

so fantastic!

how about a matrix times another matrix?

like

[?][121022041]=[121022005]\begin {bmatrix} ? \end {bmatrix} \begin {bmatrix} 1&2&1\\ 0&2&-2\\ 0&4&1 \end {bmatrix} =\begin {bmatrix} 1&2&1\\ 0&2&-2\\ 0&0&5 \end {bmatrix}

maybe you have thought out.

[100010021][121022041]=[121022005]\begin {bmatrix} 1&0&0\\ 0&1&0\\ 0&-2&1 \end {bmatrix} \begin {bmatrix} 1&2&1\\ 0&2&-2\\ 0&4&1 \end {bmatrix} =\begin {bmatrix} 1&2&1\\ 0&2&-2\\ 0&0&5 \end {bmatrix}

文章目录
  1. 1. 01#The Geometry of Linear Equations
    1. 1.1. a tip and enlightenment
    2. 1.2. elimination
  2. 2. 02#Elimination with Matrices