Show that the following are not subspaces of R^m

Question

Can I get some guidance on this question?

Thank you

Answer 1

HINT

Recall that by definition the zero vector must belong to any subspace.

Answer 2

A subspace must contain the sum of two vectors belonging to it, but the sum of two vectors of the type in part (a) cannot have 2 as a first component.

0 must belong to any subspace of a vector space.

As for three, if the sum of the components of vector $v$ and the sum of components of vector $w$ both are 1, the sum of the components of vector $v+w$ is not 1.

Try to get more acquainted with the definitions.

Answer 3

Given a vector space $V$ and its subspace $W$, considering any $v\in V, w\in W$:

1. First, notice that $0\cdot v=\mathbf{0_V}$, which can be proved:
   $0\cdot v+ 0\cdot v=_{\textrm{by axiom 8}} (0+0)\cdot v=0\cdot v$, which means $0\cdot v=\mathbf{0_V}$ by cancellation law, which can also be proved using those axioms.

2. Second, notice that $0\cdot w=\mathbf{0_V}$, since $W$ is just a subset of $V$, so $w\in W$ also means $w\in V$, by (1.) we have this equality hold.

3. Finally, since $w+\mathbf{0_W}=w$, but for the same reason $w\in V$, so $w+\mathbf{0_V}=w.$ by cancellation law, we have that $\mathbf{0_W}=\mathbf{0_V}=_{\textrm{as a shorthand}}\mathbf{0}$.

So this is the story of why anyone here tell you to check whether $\mathbf{0}$, which means $\mathbf{0_V}$, is in the subspace under the discussion.

Anyway, don't be discouraged by the downvotes. But try to think about why people don't consider it a good question!