For one of my homework assignments, the question posed is as follows:
Determine if the linear systems $A \vec x = \vec 0$ and $B \vec x = \vec 0$ are equivalent where:
To solve, I would assume you attempt to reduce until A and B have the same row echelon form, if impossible we could conclude that the two are not equivalent (right?). However, I'm a little confused as to what is meant/what it entails to have $A \vec x = \vec 0$ and $B \vec x = \vec 0$. Any help phrasing the question in more explicit terms? Cheers.
$A\overrightarrow{x} = \overrightarrow{0}$ and $B\overrightarrow{x} = \overrightarrow{0}$ should be considered with the condition that these statements hold for all values of $x$ in the domain of $A$ and $B$.
What do you know about vectors in the domain which get mapped to $0$, but are not $0$ themselves? Their span makes up the nullspace of their operator. So if $\mathscr{N}(A) \ne \mathscr{N}(B)$, then they cannot be equivalent, simply because $A\overrightarrow{x} = B\overrightarrow{x}$ will not hold for all values in the respective nullspaces of $A$ and $B$.
Identify the vectors spanning the nullspace of each operator, then show that they do not span the same space, or that they do.