Vector spaces and subspaces

Column space of A solving Ax=b

Null space of A

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

All combs cv+dw are in the space

向量空間對數乘和加法需要封閉

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, 原因在于對加法不封閉，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, 交點為zero

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Column space of A（列空間），記作C（A）

example:

???? is a subspace of R^4, 記作 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)

接下來考慮nullspace of A: all solutions to Ax=0

now write some solutions, such as

觀察規律可總結出一般形式

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,即對加法和數乘都封閉

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another example：

解中不包含zero vector，故不構成space，那么它的解是什么樣的呢？

是不穿過原點的平面或直線

summary：

subspace：1、combination of several vectors

?????????????????????? 2、從方程組中通過讓x滿足特定條件

轉載于:https://www.cnblogs.com/nanocare/p/6015487.html

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