Subspaces

Subspaces are another very important concept that state that we can have one or many vector spaces inside another vector space. Let's suppose V is a vector space, and we have a subspace . Then, S can only be a subspace if it follows the three rules, stated as follows:

• 
• and , which implies that S is closed under addition
•  and  so that , which implies that S is closed under scalar multiplication

If , then their sum is , where the result is also a subspace of V.

The dimension of the sum  is as follows: