# N linear equations, N unknows – 方程式组和矩阵

begin{cases} 2x-y=0 -x+y=3 end{cases}

begin{bmatrix} 2&-1 \ -1&2 end{bmatrix} * begin{bmatrix} x \ y end{bmatrix} = begin{bmatrix} 0 \ 3 end{bmatrix}

A matrix is just a rectangular array of numbers.

# Row Picture

Row Picture, 也就是从行的角度来考虑矩阵内在的含义.

# Column Picture

x begin{bmatrix} 2 \ -1 end{bmatrix} + y begin{bmatrix} -1 \ 2 end{bmatrix} = begin{bmatrix} 0 \ 3 end{bmatrix}

2x-y=0 \ -x+2y-z=-1 \ -3y+4z=4

A=begin{bmatrix}2&-1&0-1&2&-1&-3&4end{bmatrix} b=begin{bmatrix}0-14end{bmatrix} )

x begin{bmatrix}2 \ -1 \ 0 end{bmatrix} + ybegin{bmatrix}-1\2-3end{bmatrix}+zbegin{bmatrix}0-1\4end{bmatrix}=begin{bmatrix}0-1\4end{bmatrix}

begin{bmatrix}1\1-3end{bmatrix}, 则可以得出一个解为x=1,y=1,z=0

Can I solve Ax=b for every b ?
Or
Do the linear combinations of the columns fill three dimensional space?

# Matrix Form – 矩阵形式

e.g.
begin{bmatrix}2&5\1&3end{bmatrix}*begin{bmatrix}1\2end{bmatrix}

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

begin{bmatrix}2&5\1&3end{bmatrix}_begin{bmatrix}1\2end{bmatrix} =begin{bmatrix}2_1+5_2\1_1+3*2end{bmatrix} =begin{bmatrix}12\7end{bmatrix}

Ax is a combination of the columns of A .