Section 2.7 PA=LU
and Section 3.1 Vector Spaces and Subspaces
Transpose(转置）
example:
特殊情况，对称矩阵（symmetric matrices),例如：
思考：R^R(R的转置乘以R)有什么特殊的？
回答：always symmetric
why?
Permutation（置换）
P=execute row exchanges
之前A=LU是建立在no row exchanges 的基础上的，但不可能每一个矩阵都是完美的，有些矩阵需要通过行变换处理，
即PA=LU （any invertible A）
P= indentity matrix with reordered rows
置换矩阵是重新排列了的单位矩阵
counts reorderings(counts all the n * n permutations : n!
性质：
Vector Spaces
Example:
R^2= all 2 dimensional real vectors = “x-y”plane,
R^3= all vectors with 3 component
R^n = all vectors with n component
思考：not a vector space? what’s the condition?
回答：向量空间必须对数乘和加法两种运算是封闭的（线性组合封闭）
比如说，二维平面子空间 line in R^2 through zero vector
总结：
subspaces of R^2: all of R^2(itself), any line through zero vector (L), zero vector only (Z)
subspaces of R^3: all of R^3(itself), any plane through zero vector (P), any line through zero vector (L), zero vector only (Z)
example:
cols in R^3, all their combinations form a subspace called column space, C(A)