Inverses

Functions

• $T:V\to W$ linear. A function $U:W\to V$ is the inverse of $T$ if $TU=I_w$ and $UT=I_v$
• If $T$ has an inverse, $T$ is said to be invertible
• $T$ is invertible $⟺$ the inverse of $T$ is unique and is denoted $T^{-1}$
• $T$ is invertible $⟺$ $[T{]}_{\beta }^{\gamma }$ is invertible
  • Furthermore, $[T^{-1}{]}_{\gamma }^{\beta }=([T{]}_{\beta }^{\gamma }{)}^{-1}$

Facts

$T,U$ invertible

• $(TU{)}^{-1}=U^{-1}{T}^{-1}$
• $(T^{-1}{)}^{-1}=T$ (this means inverses are invertible)
• A function is invertible iff it is one-to-one and onto $T:V\to W$ linear and invertible
• $T^{-1}$ is also linear
• $V$ finite dimensional iff $W$ is finite dimensional and $\dim (V)=\dim (W)$

Matrices

• $A\in M_{n\times n}$ is invertible if $\exists !B\in M_{nxn}$ s.t. $AB=BA=I$
  • Note: all inverses are unique
• Every invertible matrix is a product of invertible matrices
• Finding
  • Can transform $(A|I_n)$ into $(I_n)|A^{-1}$ using finite number of elementary row operations
    • If not invertible, with product a row with zeros in first $n$ entries
      • That is because if there is a row of zeros $\text{rank}(L_A)<\dim (V)$