I was unable to find a simple proof that a subspace is a vector space. I know that a subspace $S$ is a subset of a vector space, such that:
$$\vec 0 \in S\\\vec a + \vec b \in S\\\alpha\vec a \in S$$
For the first case, I know that $\alpha = -1 \implies -\vec a \in S$
But how can we prove that $\vec a + \vec b = \vec b + \vec a$? We know that $S$ is a subset of a vector space, so the laws of the space are valid (it should be valid, then, the associativity law for $a$ and $b$). But what I don't understand, is why we can simply check for $\vec 0 \in S,\vec a + \vec b \in S,\alpha\vec a \in S$ and then conclude this subspace is also a vector space. We don't know yet, so we can't assume the laws of vector space are valid.
Best Answer
You already know $\vec a+\vec b=\vec b+\vec a$ when you calculate the two sides in the larger vector space. But that is exactly what you do when you calculate sums in the subspace too -- the sums are the same no matter whether you think of the elements as living in the subspace or in the original space.
Because equality holds in the original space, it must hold in the subspace too.