linear algebra letcure 9 & 10 08/16/2017

lecture 9:
independence:
vector x1,x2,....xn are independent, if no combination gives 0 vector (except the 0 combinations.)

c1x1+c2x2+...+cnxn is not 0, they are independent

vectors v1,...vl span a space means: the space consists of all combinations of those vectors.

Basis for a space is a set of vectors with 2 properties:

1. they are independent.
2. they span the space

Example :
space is R3
one basis: [1;0;0] [0;1;0] [0;0;1]

n vectors give basis if the n*n matrix is invertible.
(n个vector的一种组合可以得到单元矩阵，也就是单元基basis)

every basis for the space has the same number of vectors.
It is called the dimension of the space.

rank(A) = #pivot column = dimension of the column space

dimension of null space is the number of free variables. = n - r

lecture 10:

4 subspace of A(m*n)
C(A) column space ;Rm
N(A) null space ;Rn
R(A) row space = all combination of columns of At C(At) ;Rn
N(At) :null space of At. The left null space of A ;Rm


basis? dimension?

C(A)
dimension of C(A) = r (rank of the matrix)
basis: is the pivot columns of A

R(A)

dimension of R(A) = r

N(A)
basis: special solutions
dimension: n-r

N(At)
basis: special solutions
dimension: m - r
why call it left null space?
At*y = 0 => yt*A = 0t =>左零矩阵

[A,I]=>[R,E]

如何计算y?
E*A = R
从E的最后一行找，因为E*A最后会得到一个最后一行为0的矩阵