Column space and Null space...

Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary.

\begin{bmatrix} c-6d \\d \\c \end{bmatrix}

: c, d are real numbers

I want to show that W is a vector space by showing that it is a null space and a column space (I know that I do not need to show both of them...one is enough...but I just want you to check if my answer is correct...and I also have a question about columns spaces).

My answer:

For null space:

c

\begin{bmatrix} 1 \\0 \\1 \end{bmatrix}

+ d

\begin{bmatrix} -6\\1 \\0 \end{bmatrix}

So we can clearly se that is is a null space, and therefore a vector space

For column space:
c

\begin{bmatrix} 1 \\0 \\1 \end{bmatrix}

+ d

\begin{bmatrix} -6\\1 \\0 \end{bmatrix}

A =

\begin{bmatrix} 1 & -6 \\0 & 1 \\1 & 0 \end{bmatrix}

Ok...so after I get the matrix "A", how will I know if Col (A) = W or not?

(edited 13 years ago)

Reply 1

kfkle

Why not just show that your set is a span of two vectors?

Reply 2

Artus

OP

Original post by kfkle

Why not just show that your set is a span of two vectors?

I know there are many ways to do this...I'm just asking if my answer is correct...

Reply 3

kfkle

Original post by Artus

I know there are many ways to do this...I'm just asking if my answer is correct...

Technically speaking, to show that your set is a null space you need to provide a matrix whose null space is your set. Which is why this is the wrong way to go about it.

(edited 13 years ago)

Reply 4

Artus

OP

Original post by kfkle

Technically speaking, to show that your set is a null space you need to provide a matrix whose null space is your set. Which is why this is the wrong way to go about it.

Why? Is it enough if I just write the span?

Reply 5

kfkle

Original post by Artus

Why? Is it enough if I just write the span?

Yes. If you write your set as a span, and you have a theorem that tells you that all spans are vector spaces, then that's all you need to do.

Reply 6

Artus

OP

Original post by kfkle

Yes. If you write your set as a span, and you have a theorem that tells you that all spans are vector spaces, then that's all you need to do.

Ok thanks...what about the column space? Is my answer right? Also, after I find matrix "A", how can I know if shows a column space?

Reply 7

kfkle

Original post by Artus

Ok thanks...what about the column space? Is my answer right? Also, after I find matrix "A", how can I know if shows a column space?

Yes, what you did is in principle correct, but really over complicates things. You only find column spaces if you have a matrix already, in which case the column space is the span of the columns.

The problem with going in the other direction is that the span is just more fundamental than a column space. Does that make sense?

Reply 8

Artus

OP

Original post by kfkle

Yes, what you did is in principle correct, but really over complicates things. You only find column spaces if you have a matrix already, in which case the column space is the span of the columns.

The problem with going in the other direction is that the span is just more fundamental than a column space. Does that make sense?

I'm sorry, but can you clarify what you mean by this?

"You only find column spaces if you have a matrix already, in which case the column space is the span of the columns. The problem with going in the other direction is that the span is just more fundamental than a column space. Does that make sense?"

Column space and Null space...

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