How do I examine if C is a linear block code?

Question

The block code $C\subseteq GF(2) ^6$ consists of all binary sequences of length $6$ with hamming weight $w = 3$. How do I examine if $C$ is a linear block code?

Definition:

A binary code is linear if the following condition is satisfied:

$\forall c_1 \in C$, $\forall c_2 \in C$ $\Rightarrow c_1+c_2 \in C$

But I still don't understand how to show it?

Answer 1

Hint:

A linear code must contain an all-zero codeword ($c_1 + c_1 = \mathbf{0}$). What is the Hamming weight of such a codeword?