I find that subspaces of vectors makes sense, but I'm having trouble with the following problem involving a function:
Let $a$ be a real number and consider the following subset of $C[-1,1]$: $$F := \{f(x) \in C[-1,1]: f(0) = a\} $$ For which values of $a$ is $F$ a subspace of $C[-1,1]$? If $F$ is a subspace, prove it.
I am really confused on how to solve this. I understand that with vectors, one criteria for being a subspace is containing the zero vector. Would this be the same as this function needing to go through the origin so that $f(0)=0$? Thank you!
The zero vector of your space is the function $x \mapsto 0$. We know that it must be contained in that set in order for it to be a subspace. And since this set contains functions $f$ only such that $f(0) = a$ we have that $a$ must be $0$ in order for it to be a candidate subspace.
Now is it really a subspace? Yes because not only the zero vector is there but also for all vectors $f,g$ and any scalar $\lambda$ we have $$ (f + g)(0) = f(0) + g(0) = 0 \implies f+g \in F$$ $$ (\lambda f)(0) = \lambda f(0) = 0 \implies \lambda f \in F $$
Hence $F$ is subspace only if $a=0$.