Subspace

A Subspace $W$ is a Subset of a Vector Space that has:

• Addition
• Multiplication Which follows the 3 axioms referred to as the Subspace Test

1. $\vec{0}\in S$
2. S is Closed Under Addition ($\vec{x},\vec{y}\in W⟹\vec{x}+\vec{y}\in W$ )
3. S is Closed Under Scalar Multiplication ($\vec{x}\in S,k\in R⟹k\vec{x}\in W$) Axiom 2 and 3 can be grouped to say that subspaces are Closed Under Linear Combinations ($x,y\in W⟺\forall c\in F,cx+y\in W$)

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 $\forall c\in R,cx+y\in W$

Proving Subspace

1. Prove all 3 axioms for the subset $S$ Or
2. Prove that $S$ is a Spanning Set for another set $X$ Proving Subspace with Spanning Sets

Concepts

• Proving Subspace Theorem
• Subspace Intersection Theorem
• Sets for R^n are Subspaces for R^n Corrolary
• Represent Subspace as Span

Subspace Operations

• Sum of Subspaces
• Subspace Addition Span Union
• Intersection of Subspaces