Subspaces

A subset, $W$, of a vector space, $V$, over a field, $F$, is a subspace of $V$ if $W$ is a vector space over F with addition and scalar multiplication defined on V

Conditions

1. $x+y\in W$ whenever $x,y\in W$
2. $cx\in W$ whenever $c\in F,x\in W$
3. $W$ has a zero vector
4. Each vector in $W$ has an additive inverse in $W$

You don’t actually need 4 because it follows from the other three (Theorem 1.3)

Examples

• All symmetric Matrices in $M_{n\times n}$ is a subspace of $M_{n\times n}$
• $P_n(F)$ is a subspace of $P(F)$
• Set of diagonal matrices

Properties

• Any intersection of subspaces of a vector space $V$ is a subspace of $V$ (theorem 1.4)
• A union of subspaces of $V$ is generally not a subspace of $V$