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

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.

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:

• 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.