Prove the set of all vectors in $\mathbb{Z}^n_2$ with an even number of 1's, over $\mathbb{Z}_2$ with the usual vector addition and scalar multiplication, is a vector space.
I am not sure how to prove closure under addition, nor am I sure how to formally prove closure under scalar multiplication.
Clearly it is closed under scalar multiplication as the only scalars are 0 and 1, which when multiplied will remain in $\mathbb{Z}^n_2$.
Not sure where to start with closure under addition.