Equivalence Relation? Column-equivalence on the set of all $m\times n$ matrices.

Question

How do I show that column equivalence on the set of all $m\times n$ matrices is an equivalence relation?

I know that to show it is an equivalence relation, I need to show that column equivalence is transitive, symmetric, and reflexive.

Answer 1

Hint: the expression "column equivalence" does most of your work for you.

I know that to show it is an equivalence relation, I need to show that column equivalence is transitive, symmetric, and reflexive.

Just do exactly that: show that your column equivalence is transitive, symmetric, and reflexive on the set of all $m\times n$ matrices. To do this:

Review your notes and/or text as reference, and answer:

• What must be true for column equivalence to be reflexive? $\\$ For an arbitrary $m\times n$ matrix we'll call $A$, are the columns of $A$ equivalent to the columns of $A$?
• What must be true for column equivalence to be symmetric? $\\$ If matrix $A_{m\times n}$ is column equivalent to matrix $B_{m\times n}$, is $B$ column equivalent to $A$?
• What must be true for column equivalence to be transitive? $\\$ Likewise, does the column equivalence of $A$ with $B$ and the column equivalence of $B$ with $C$ imply that $A$ and $C$ must be column equivalent?

If you can answer "yes" to each of the above, then we've established that column equivalence on $m\times n$ matrices is an equivalence relation.

Unpacking the definitions of the properties of an equivalence relation is key here, along with the fact that the relation in question is defined as "column equivalence".