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:

$Ax$ is a combination of columns of A.

For example

A equals

$\begin {bmatrix} 2 & 5\\ 1 & 3 \end {bmatrix}$

x equals

$\begin {bmatrix} 1\\ 2 \end {bmatrix}$

then we can get

$1 \begin {bmatrix} 2\\ 1 \end{bmatrix} +2\begin{bmatrix} 5\\ 3 \end{bmatrix}= \begin{bmatrix} 12\\ 7 \end{bmatrix}$

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

$\begin {bmatrix} 1 & 2 &4 \end {bmatrix} \begin {bmatrix} 3&4&5\\ 1&-2&6\\ 0&1&2 \end {bmatrix}$

okey. we will try to solve it.

it gives

$1\begin {bmatrix} 3&4&5 \end {bmatrix} +2\begin {bmatrix} 1&-2&6 \end {bmatrix} +4\begin {bmatrix} 0&1&2 \end {bmatrix} =\begin {bmatrix} 5&4&25 \end {bmatrix}$

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

so fantastic!

how about a matrix times another matrix?

like

$\begin {bmatrix} ? \end {bmatrix} \begin {bmatrix} 1&2&1\\ 0&2&-2\\ 0&4&1 \end {bmatrix} =\begin {bmatrix} 1&2&1\\ 0&2&-2\\ 0&0&5 \end {bmatrix}$

maybe you have thought out.

$\begin {bmatrix} 1&0&0\\ 0&1&0\\ 0&-2&1 \end {bmatrix} \begin {bmatrix} 1&2&1\\ 0&2&-2\\ 0&4&1 \end {bmatrix} =\begin {bmatrix} 1&2&1\\ 0&2&-2\\ 0&0&5 \end {bmatrix}$