Can a subspace of a vector space $V$ have a different form of zero vector?

Question

In Linear algebra,

a necessary condition for a subset of a vector space to be a subspace, is to contain the zero vector.

For example,

The following subspaces of Vector space $\mathbb {R^3}$ has

$0$-D subspace(the origin), 1-D subspace (lines through the origin),2-D subspace (planes through the origin) and finally $\mathbb {R^3}$ have the same zero vector

But the subspaces of Vector space of all polynomials with degree at most n, $\mathbb{P_n}$ have the same form of zero vector defined as the function value to be $0$ $\forall$ t , with different coordinate mappings which are one to one and onto $\mathbb{R^k}$ with $k$ = dim $\mathbb{P_m}$ $\forall$ $m < n$

Can a subset of a Vector space $V$ having a zero vector in its own right different from that of $V$ ( may be non trivial, here is an example Zero vector of a vector space) closed under vector addition and scalar multiplication be called a subspace of $V$ ?

Answer 1

Let $V$ be a vector space and $W$ is a subspace. Let $0_V$ be the zero of $V$ and $0_W$ be the zero of $W$.

For every $v \in V$ it holds that $v-v = 0_V$ and for every $w \in W$ it holds that $w-w = 0_W$

As $W \subseteq V$ and $W \ne \emptyset $ there is a $w\in W$ and $0_V = w-w = 0_W $

Answer 2

The zero vector of a subspace is always the same as the zero vector of the whole space.

This is simply because $0=v-v$ for any vector $v$.

So, taking any vector space $V$ and its subspace $W$, if we take any vector $w\in W$, the zero vector of $W$ is $w-w$, which is also the $0$ vector of $V$.

Answer 3

No, because a vector subspace by definition has the same operations as the original vector space, restricted to a subset. If $w+v = v$ in your subspace, then $w=w+v-v = v-v = 0$ in the original space, so a $0$ in the subspace must be $0$ in the original.