MHB Relation between null and column space

[Dethrone]

Is there a relationship between 1 and 2. If so, is it 1 implies 2, 2 implies 1, or if and only if.
1) $\operatorname{null}A=\operatorname{null}B$
2) $\operatorname{col}\operatorname{rref}A=\operatorname{col}\operatorname{rref }B$

 

[I like Serena]

Is there a relationship between 1 and 2. If so, is it 1 implies 2, 2 implies 1, or if and only if.
1) $\operatorname{null}A=\operatorname{null}B$
2) $\operatorname{col}\operatorname{rref}A=\operatorname{col}\operatorname{rref }B$

Hey Rido! ;)

Nice operatornames! (Mmm)

Actually, I'm not quite clear on what $\operatorname{col}$ is supposed to mean.
It it the column space? Or the rank of the column space?
Can you clarify?

And what about $\operatorname{null}$? Is it whether the matrix is a null matrix, or is it the kernel $\operatorname{ker}$? (Wondering)

 

[Dethrone]

Hey Rido! ;)

Nice operatornames! (Mmm)

Actually, I'm not quite clear on what $\operatorname{col}$ is supposed to mean.
It it the column space? Or the rank of the column space?
Can you clarify?

And what about $\operatorname{null}$? Is it whether the matrix is a null matrix, or is it the kernel $\operatorname{ker}$? (Wondering)

$\DeclareMathOperator{\adj}{adj}$
$\DeclareMathOperator{\null}{null}$
$\DeclareMathOperator{\col}{col}$

Hi ILS! :D

$\col A$ represents the column space of $A$, so in this case, it is the column space of the row reduced form of $A$. The matrix is the null matrix.

 

[I like Serena]

Let's pick a couple of simple matrices. (Thinking)

Say $A=(^{2\ 0}_{0\ 0})$ and $B=(^{0\ 3}_{0\ 0})$.

What are $\operatorname{null} A$ and $\operatorname{null} B$?
And $\operatorname{rref} A, \operatorname{rref} B$?
And $\operatorname{col} \operatorname{rref} A, \operatorname{col} \operatorname{rref} B$?

Do they give you a clue? (Wondering)