Linearity of codes

Question

Assuming $C$ is a binary linear code and let $a$ $\notin $ $C$ be any vector. Show that $C$ $\cup (a + C) $ is also linear.

I know that for any $C_1,C_2 \in C $ then $\alpha C_1 + \beta C_2 \in C$ but now how do i apply this to this situation? Is it even the right way to go about this problem? All suggestions and ideas are deeply appreciated. thanks

Answer 1

In brief, Considering the cases clearly stipulated by @Git Gud, First taking $C_1, C_2 \in C$ then we have

$\alpha C_1 + \beta C_2 \cup a + \alpha C_1 + \beta C_2 \forall \alpha,\beta\in { 0,1} $.

Clearly the left part of the $ \cup$ sign is $\in C $ and the right part is $\in a + C $

Next, taking $C_1,C_2 \in a + C$ it follows that;

$C \cup \alpha (a + C_1 + C_2 ) + \beta (a + C_1 + C_2 ) $

the right part of the $\cup$ is in $ a + C $ hence the union holds $ \forall \alpha , \beta \in 0,1 $

finally, taking $ C_1 \in C,C_2 \in a + C $ we get;

$\alpha C_1 + \beta (a + C_2) \in C \cup (a + C) $ definitely and we are done.