# Linear Algebra: equivalence classes

Give the corresponding equivalence classes for $3\times2$ matrices over the field $\mathbb{Z}_5$ for row and column equivalence classes.

Whenever $B$ can be obtained from $A$ by performing a sequence of column operations, we write $A$ ~ $cB$, and we say that $A$ and $B$ are column equivalent. In other words $A$~$cB$ if and only if, $AQ=B$, where $F=F_1F_2F_3$ and each $F_i$, $1\leq i\leq l$ are elementary column matrices. Whenever $B$ can be derived from $A$ by a combination of elementary row and column operations we write $A$~$B$, and we say $A$ and $B$ are equivalent matrices. $A$~$B$ if and only if $PAQ=B$ for $P=E_kE_{k-1}\cdots E1$ and $Q=F_1F_2\cdots Fl$

Im not sure hot to get started on this problem, if someone would please point me in the right direction and maybe walk with me through a few steps that would be great.

-
Thank you for clarifying. (I am guessing that row equivalence is defined as on Wikipedia.) – Jonas Meyer Jan 21 '13 at 22:44
@user5454 You need to define the relation on the question. We don't know what it is. At least I don't. – Git Gud Jan 21 '13 at 22:47
Using just row operations, you can bring any matrix to reduced row-echelon form. There are only a few reduced row-echelon forms for $3\times2$ matrices over the field of $5$ elements. Then you can use column operations on those forms to get down to a complete set of row-column inequivalent matrices. – Gerry Myerson Jan 21 '13 at 23:06