Matrix algebra over finite fields

[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!

 

[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.

 

[tacman]

Hi there,

Matrix algebra over finite fields, also known as Galois fields, is a very interesting and useful topic in mathematics and engineering. In order to determine if the rows of a matrix over GF(p) are linearly independent, you can use the following steps:

1. Write the matrix in reduced row echelon form using Gaussian elimination. This will help you to spot any patterns or relationships between the rows.

2. Look at the leading entries in each row. If they are all different, then the rows are linearly independent. If there are any rows with all zeros, then those rows are linearly dependent on the other rows.

3. If there are any duplicate rows in the reduced row echelon form, then the rows are linearly dependent.

To determine if two nonidentical matrices have the same row space, you can follow these steps:

1. Write both matrices in reduced row echelon form.

2. Compare the leading entries in each row. If they are the same in both matrices, then the rows are in the same row space. If there are any missing leading entries in one matrix, then the row space of that matrix is a subspace of the row space of the other matrix.

3. If the reduced row echelon forms are different, then the matrices have different row spaces.

I hope this helps! If you have any further questions or need clarification, please don't hesitate to ask. Good luck with your studies!