Hi I need to show that
$$C_1 \subseteq C_2 \Leftrightarrow C_2^{⊥} \subseteq C_1^{\perp}$$
In guess I need to use standard form matrices for generator matrix and parity check matrix(also parity check matrix of a code is generator matrix of its dual code). C1 should have less words than C2, then more words will be orthogonal to C1. But how to put this together?
$ x \in C_2^\perp \Leftrightarrow \forall y \in C_2, \langle x, y\rangle = 0\\ \Rightarrow \forall y \in C_1 \langle x,y\rangle = 0 \\ \Rightarrow x \in C_1^\perp$