1.
Think back to your linear algebra course. With your group, give definitions for the following terms: span, linear independence, basis, dimension, row space.
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Worksheet Linear Algebra with Finite Fields
Throughout this activity, we will refamiliarize ourselves with key linear algebra concepts, but this time working over finite fields such as \(\F_2\) and more generally \(\F_p\) instead of the real numbers.
2.
Which, if any, of the definitions you provided in the previous problem do you think need to change when working over a finite field such as \(\F_p\) instead of the real numbers? Explain.
3.
Try applying your definitions to the following set of vectors over \(\F_7\text{.}\) (For basis and dimension, consider the vector space they span; for row space, consider the matrix with these vectors as rows)
\begin{gather*}
S=\{ (1,2,3,4,5), (1,0,1,2,1),(0,2,1,0,4),(2,3,4,5,1) \}
\end{gather*}
4.
Give two different generator matrices for the subspace generated by the set \(S\text{.}\)
5.
Give a parity-check matrix for the subspace generated by the set \(S\text{.}\)
6.
How many vectors are in a subspace of dimension \(k\) over the finite field \(\F_q\text{?}\) Justify your answer.
7.
How many ordered bases are there for a subspace of dimension \(k\) over the finite field \(\F_q\text{?}\) Justify your answer.
8.
How many \(k\)-dimensional subspaces are there of \(\F_q^n\text{?}\) Justify your answer. This number is called the “Gaussian binomial coefficient” or “\(q\)-binomial coefficient” and is often denoted
\begin{equation*}
\begin{bmatrix} n \\ k \end{bmatrix}_q
\end{equation*}
