MATH1012 - Lecture 13

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Skeleton Slides

Change of Basis

• 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 $f$

Eigenvectors

• If $A$ is an $n\times n$ matrix, then a non-zero vector $v$ is called an eigenvector of $A$ with eigenvalue $\lambda$, if $Av=\lambda v$
  • i.e. multiplying by $A$ results in a scalar multiplication of $v$
• Eigenvalues can be found/proven through row-reduction
  • A system has non-zero solutions when $\det (A-\lambda I)=0$
    • $Av=\lambda v$ → $(A-\lambda I)v=0$
    • $I$ is identity matrix
  • Solving the determinant for $\lambda$ can give us all eigenvalues

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MATH1012 - Lecture 13
Annotated Slides