Let $C \subseteq \mathbb{F}^n_q$ be a linear code with the dimension $k$ and $G$ the generating matrix of $C$, in which no row disappears.
We define a linear subspace $T:=C\ \cap \{x:x_i=0\}$ with $i\in \{1,...,n\}$.
Show: $dim\ T =k-1$
I first started with looking at $T$ and what elements are in $T$. I came to the conclusion that these are:
$T=\{(0,x_2,...,x_n)^T, (x_1,0,...,x_n)^T, ..., (x_1,x_2, ..., 0)^T\}$.
Since T is a linear subspace, then $(0,...,0)^T\in T$. The only elements that are not in $T$ are then $\{(x_1,...,x_n)^T:x_i \not= 0$ for all $i\in\{1,...,n\}\}$.
I am not sure if my conclusions are right, but if they are, I am not sure how this affects the $dim\ T$.