Linear Algebra Lecture 6

Linear Algebra Lecture 6

1.Vector spaces and subspaces

2. Column space of A

3. Null space of A

Vector space

Vector space requirements v+w and cv are in the space, all the linear combinations cv+dw are in the space.

Example 1

Take a plane P and a line L in R3 space,

Is P⋃L a subspace or not?

No, because can’t add.

Is P⋂L a subspace or not?

Yes.

Column space of A

A=⎡⎣⎢⎢⎢123411112345⎤⎦⎥⎥⎥, what is in C(A)?

A is a subspace of R4. Take 3 column’s linear combinations. The column space of A is all linear combinations of the columns.

Ax=⎡⎣⎢⎢⎢123411112345⎤⎦⎥⎥⎥⎡⎣⎢x1x2x3⎤⎦⎥=⎡⎣⎢⎢⎢b1b2b3b4⎤⎦⎥⎥⎥

Does Ax=b have a solution for every b?

No, because A has 4 equations and 3 unknowns.

The combinations of these columns don’t fill the whole four dimensional space.There’s going to be some vectors b that are not combinations of these three columns.

Which vectors b allow this system to be solved?

I can solve Ax=b exactly when the right-hand side b is a vector in the column space. Because the column space contains all the combinations, all the Ax. So those are the b*s that I can deal with. If b is not a combination of the columns, then there is no *x, there is no way to solve Ax=b.

Are those three columns independent(线性无关)?

No, because column 3 is the sum of column 1 and 2. So these two columns are pivot columns(主列). The column space is a two dimensional subspace of R4.

Null space of A

Null space of A contains all solutions x, to the equation Ax=0.

Ax=⎡⎣⎢⎢⎢123411112345⎤⎦⎥⎥⎥⎡⎣⎢x1x2x3⎤⎦⎥=⎡⎣⎢⎢⎢0000⎤⎦⎥⎥⎥, what is in N(A)?

c⎡⎣⎢11−1⎤⎦⎥

The solutions to Ax=0 always give a subspace?

If Av=0 and Aw=0, then A(v+w)=0, then cAv=0, so the null space is always a vector space.