Linear Map Inverse

Definition

• Suppose that $S:V\to W$ is a function
• A inverse of $S$ is $T:W\to V$ s.t
  • $\forall v\in V,TS(v)=v$
  • $\forall w\in W,ST(w)=w$

Uniqueness Definition

$T$ invertible $⟺\forall b,Tv=b$ has a Unique Solution

Theorems

• Injectivity Surjectivity and Inverses
• Determinants and Invertiblity Theorem
• Invertibility and Eigenvalues
• Linear Map Inverses are Unique
• Linear Map Inverses are Linear