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 operations, is a vector space.

Question

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.

Answer 1

Show that it's the kernel of the linear functional: $$f:(\mathbb Z/2\mathbb Z)^n\rightarrow\mathbb Z/2\mathbb Z$$ $$f(a_1, a_2, \cdots, a_n) = a_1+a_2+\cdots+a_n$$