1. Vector space
Vector space requirements++v+w++ and ++c v++ are in the space, all combs ++c v + d w++ are in the space
但是"子空间"和"子集"的概念有区别，所有元素都在原空间之内就可称之为子集，但是要满足对线性运算封闭的子集才能成为子空间
中 2 subspaces
L: line is a subspace
P: Plane through 0,0,0T is a subspace of
= all vectors in P or L or both is not a subspace
= all vectors in both P and L is a subspace - null space
2. 列空间 Column space
column space of A is subspace of 
is C(A)=all linear combs. of columns
Does Ax=b have a solution for every b? No
cuz 4 equations and 3 unknowns 列向量的线性组合无法充满
which b's allow this system to be solved?
Can solve Ax=b exactly when b is in C(A) IN
由于列向量不是线性无关的，第三个列向量为前两个列向量之和，所以尽管有3个列向量，但是只有2个对张成向量空间有贡献。矩阵A 的列空间为内的一个二维子空间
3.零空间（或化零空间）Nullspace
Null space of A = all solutions x = 
in to Ax=0
对于所给定这个矩阵A ，其列向量含有4个分量，因此列空间是空间的子空间。
x 为含有3个分量的向量，故矩阵A 的零空间是的子空间。对于mxn矩阵，列空间为
的子空间，零空间为
空间的子空间。
N(A) contains 
which is a line in
check that- solution to Ax=0 always give a subspace
if Av=0 and Aw = 0 then A(v+w)=0
then A(12v)=0
4. influence of b
subspaces have to go through the origin
5. summary:
2种构筑子空间方法
1.对于列空间，它是由列向量进行线性组合张成的空间
2.零空间是从方程组出发，通过让x满足特定条件而得到的子空间