Linear Algebra lecture6 note

Vector spaces and subspaces

Column space of A solving Ax=b

Null space of A

 


Vector space requirements v+w and cv are in the space

All combs cv+dw are in the space

向量空间对数乘和加法需要封闭

subspace of R^3:

Line( L) through zero vector  is a subspace of R^3

Plane( P) through zero vector is a subspace of R^3

then we got 2 subspaces: P and L

P∪L means all vectors in P or L or both, this is not a subspace, 原因在于对加法不封闭,加和后所得的可能既不在P上,也不在L上

P∩L means all vectors in both P and L, this is a subspace, 交点为zero

 


Column space of A(列空间),记作C(A)

example:

Linear Algebra lecture6 note     is a subspace of R^4, 记作 C ( A)

思考:Does Ax=b have a solution for every b? Which b’s allow this system to be solved?

回答:No. 4 equations, 3 unknowns, we can solve Ax=b exactly when b is in C( A)

Linear Algebra lecture6 note

接下来考虑nullspace of A: all solutions to Ax=0

Linear Algebra lecture6 note

now write some solutions, such as

Linear Algebra lecture6 note

观察规律可总结出一般形式

Check the solution to Ax=0 always give a subspace

If Av=0 and Aw=0, then A(v+w)=0, then A(12v)=0,即对加法和数乘都封闭

 

another example:

Linear Algebra lecture6 note

解中不包含zero vector,故不构成space,那么它的解是什么样的呢?

Linear Algebra lecture6 note是不穿过原点的平面或直线

summary:

subspace:1、combination of several vectors

                       2、从方程组中通过让x满足特定条件

上一篇:Jenkins系列之二——centos 6.9 + JenKins 安装


下一篇:SQL主、外键,子查询