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Matrix algebra over finite fields

  1. Dec 11, 2007 #1

    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. jcsd
  3. Dec 11, 2007 #2


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    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.
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