A：the matrix of coefficients

The vector of unknowns,stand for X

The right-hand side is also a vector that call b

linear equations are Ax=b

Row picture:

column way:

The equation of matrix called linear combination of columns

column picture:

The x[2 -1] can be the vector of col 1.

The y[-1 2] can be the vector of col 2.

so we need to find the value of x and y,that can make the final vector.

By the answer we got the value are x=1 and y=2.

so the [2 -1] *1 in the picture is stay in the same position

but the [-1 2]*2 in the picture is extend twice.

Then

Three equations and three unknowns:

The three planes cross the main point by they intersection.but it is hard to see the main point on the graph.

so in the three dimensional picture,row picture can not be worked.

linear combination of 3D

因?yàn)槭翘氐卦O(shè)定好的題目，所以xyz value are easy to find.

Hence the final vector of the 3-D linear combination is col 3 that is b [0-1 4]

sometimes we could change the value of b and make the left vector match the equation coincidentally.

Q 1:can i solve Ax=b for every b?

Ans 1:yes.

Q 2:Do the linear combination of the columns fill 3-d space?

Ans 2:yes for the Q in this video.

Through the video,we can know the matrix have kinds,non-singular and invertible matrix.

when the b is not in the same plane by others column. so that called singular case.The matrix would be not invertible.

chapter 2:消元法[Elimination]

Elimination:success&failure

It's the way every software package solves equations

Tree equations and tree unknowns can use elimination method to solve

use the matrix operations(矩陣變換)

under the elimination of equations,so the pivot(主元)is the main point to eliminate the equations.

1 is the first pivot,so the 1 2 1 are the pivot row. It does not changed.

pivot can not be 0

if the pivot be 0,so the matrix is invertible

New concept:(增廣矩陣)augmented matrix in the Back-substitution