Change of Basis

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Introduction

• Let $B$ and $C$ be two bases for the same $m$-dimensional subspace; $V$
• The transition matrix $P_{CB}$ is given by $(v{)}_C=P_{CB}(v{)}_B$
  • Recreate each vector in $B$ using the vectors in $C$ to find the transition matrix (example in annotated slides)
  • $P_{CB}$ takes $B$-coordinate vector and produces $C$-coordinate vector
• The inverse of $P_{CB}$ equals $P_{BC}$ ($(P_{CB}{)}^{-1}$ = $P_{BC}$)
  • Additionally $P_{AB}[f{]}_{BB}{P}_{BC}=[f{]}_{AC}$
  • $[f]$ is the standard matrix of a linear transformation $f$

Change of Basis Matrix

• $P_{CB}$ is the change of basis matrix
  • It takes a vector in $B$ coordinates and outputs it in $C$ coordinates
• $[f{]}_{CB}$ is the linear transformation $f$ when the input is in $B$ coordinates and the output is in $C$ coordinates
  • It takes a vector in $B$ coordinates, applies the function $f$ and then outputs the answer in $C$ coordinates