Def of Row vector and column vector:
For any m-by-n matrix
, the vectors , , , in that are formed from the rows of are called the row vectors of , and the vectors , , , in formed from the column of are called the column vectors of .
Def of row space, column space and null space:
If
is a m-by-n matrix, then the subspace of spanned by row vectors of is called the row space of , and the subspace of spanned by the column vectors of is called the column space of . The solution set of , which is a subspace of , is called the null space of . Row Space(rsp) of
Column Space(csp) of
Null Space of
（The solution set of ） Let
be a m-by-n matrix, then
Lemma(矩陣column向量空間等於其像集):
Let
be a m-by-n matrix then
proof of Lemma:
(By (1))
Theorem(由csp判斷線性方程是否有解):
A system of linear equations
is consistent if and only if is in the column space of .
若有解意味著 定在A的像集中，由上Lemma ，b定在 中 >
Proof:
Let
be a m-by-n matrix and is consistent.
(by Lemma)
EX:
Theorem(求解AX=b的另一種方法):
If
is any solution of a consistent linear system and if is a basis for N(A) then the general solution of is
where are scalars