Consider the matrix $$A=\begin{pmatrix} 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 \end{pmatrix}.$$ I am trying to find a basis for $A$. From what I understand, a basis must consist of linearly independent columns that span the entire space.
Is the following set a basis: $$\left\{\begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix},\begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix}\right\}?$$
Original question
Let $C$ be the binary linear code with parity check matrix $$H=\begin{pmatrix} 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 \end{pmatrix}.$$ and let $B$ be a basis for $C$. How many codewords does $B$ contain?
Answer is $2$.