Determine whether the subset $A_1 = \{f : \mathbb R → \mathbb R : f(0) = 0\}$ of $F$ is a subspace of $F$

Question

Where $F$ is the vector space over $\mathbb{R}$ that is the set of all functions $f : \mathbb{R} → \mathbb{R}$.

Now, I'd be super happy to prove this under the zero vector, addition and scalar multiplication if the question wasn't asking for $f(0) = 0$. But I can't figure out how to prove it under addition. I can't substitute in any vectors u or v because there's a fixed function or am I missing something?

Answer 1

Let $f,g \in A_1$ and $ \alpha, \beta \in \mathbb R$. Define $h: = \alpha f + \beta g$.

Then: $h(0)= \alpha f(0) + \beta g(0)=\alpha \cdot 0 + \beta \cdot 0=0$, hence $h \in A_1$ and you are done.