Which operations does a subset of a vector space have?

Question

Let $(V, K, +, *)$ be a vector space over a field $K$. My book defines a subspace of $V$ as the subset $W \subset V$ that satisfies the following conditions:

1. If $v, w \in W$, then $v+w \in W$.
2. If $c \in K$ and $v \in W$, then $cv \in W$.
3. The element $O \in V$ is also an element of $W$.

In other words, we want the image by the operation $+$ of $V$ of two elements of $W$ to be in $W$, and the image by the operation $*$ of $V$ of a scalar and an element of $W$ to be in $W$.

Now, showing $W$ is a vector space with operations $+$ and $*$ of $V$ is straightforward; for example, lets try to show that, given $u$, $v$ and $w$ of $W$, we have $(u+v)+w = u+(v+w)$: since $W \subset V$, all $u$, $v$ and $w$ belong to $V$. $+$ of the original vector space satisfies this property. $\square$

I'm wondering about if we actually can say that $W$ is a vector space with the operations $+$ and $*$ of $V$. Or better, are the operations $+$ and $*$ of the subspace actually the same operations $+$ and $*$ of the original vector space $V$?

Answer 1

Yes, the operations $+$ and $∗$ of the subspace $W$ the same operations $+$ and $∗$ of the original vector space $V$ by definition of subspace.

Answer 2

Yes, this is the main idea: all operations and standard properties of the larger vector space are preserved. We should only show, that a subspace is closed under linear combinations.