Proving Subspace Theorem

Theorem

1. $W$ is a non-empty Subset of vector space $V$
2. $W$ is a subspace of $V⟺\forall x,y\in W$ and $\forall c\in R,cx+y\in W$

Proof

Assume W is a non-empty subset of vector space V

$⟹$

1. Assume W is a subspace of V
2. Then, $\forall x\in W,\forall c\in R,cx\in W$
3. Then, $\forall y\in W,cx+y\in W$ as well
4. By definition of subspace, $W$ of $V$ is Closed Under Addition and Closed Under Scalar Multiplication.

$⟸$

1. Suppose $W$ is a subset of $V$ that satisfies $\forall c\in R,cx+y\in W$
2. Take $c=1$
   1. $x+y\in W$ thus, $W$ is Closed Under Addition
3. Take $x=y$ and $c=-1$
   1. Then, $-y+y=0\in W$
4. Take $y=0$
   1. Then $cx\in W$, thus $W$ is Closed Under Scalar Multiplication
5. Thus, $W$ is a subspace $\square$