Let F be a field and let V=F^F, which is a vector space over F. Let w be the set of all functions f element of V satisfying f(1)=f(-1). Is W a subspace of V?
a. Has the zero vector b. closed under vector addition. c. closed under scalar multiplication.
can someone show me how to prove this?
Let the map
$$\Phi\colon V\rightarrow \Bbb F, \quad f\mapsto f(1)-f(-1)$$ the we see easily that $\Phi$ is a linear form: $$\Phi(\lambda f+g)=\lambda \Phi(f)+\Phi(g)$$ so $W=\ker \Phi$ is an hyperplane (subspace) of $V$.