Linear codes and check matrices. Two linear codes are defined as the following subspaces of $\Bbb B_7$:
$C_1$ with dimension $2$ and basis $\{1010101,\ 0101010\}$
$C_2=\text{ker}(H)$ where $H=\left (\!\begin {array}{ccccccc} 0&0&0&1&1&1&1\\ 0&1&1&0&0&1&1\\ 1&0&1&0&1&0&1 \end {array}\!\right )$.
Just wanted to check, to find $C_1$ I'd take the word $0000000$ and add the two elements in the basis right? For $C_2$ however, I get a bit bogged down in trying to get $(A\vert I_m$) because I'm not sure what $m$ should be in this case.

$C_1$ has dimension 2 and consists of the zero vector, the two basis vectors and the all-1 vector (by adding the two basis vectors).
$C_2$ has dimension $m=4 = 7-3$, since the kernel has dimension 3 (as can be seen from the check matrix) and the ambient space is $\Bbb Z_2^7$.
