# Matrix algebra over finite fields

1. Dec 11, 2007

### AngelofMusic

Hi,

We recently started analyzing linear machines using matrix algebra. Unfortunately, I haven't had much exposure to operating in finite fields aside from the extreme basics (i.e. the definitions of GF(P)). I can get matrix multiplication/addition, etc. just fine, but it's when finding the properties of a matrix that I'm confused.

How do we know if the rows of a matrix over GF(p) are linearly independent?

More specifically, how can I tell if two nonidentical matrices have the same row space, or if the row space of matrix A is a subspace of the row space of matrix B?

I suspect the answer to my first question is just to do Gaussian elimination and look at the rank instead of doing any algebraic manipulation such as (c1*row1 + c2*row2... ) and so forth.

But suppose I've got two matrices in row echelon form. How would I compare the rowspans of both matrices once I've done that?

I may be missing something very obvious, so your patience is appreciated!

2. Dec 11, 2007

### quasar987

Linear algebra works the same over any field as it does over R.

For instance, the rows of a matrix are linearly independent if and only if the determinant is different from zero.