# Linear Algebra II

## November 3, 2014

### Lecture 22

Filed under: 2014 Fall — Y.K. Lau @ 10:23 PM

A brief summary: Given a linear operator ${T:V\rightarrow V}$ on finite-dimensional vector space ${V}$.

• For any ordered bases ${E}$ and ${F}$,

$\displaystyle M_{EE}(T)$ is similar to ${M_{FF}(T)}$.

• ${\lambda}$ is an eigenvalue of ${T}$ ${\Leftrightarrow}$ ${\lambda}$ is an eigenvalue of (any) matrix representation of ${T}$ w.r.t an ordered basis.

Their eigenvectors are related via the coordinate isomorphism. (See p.458 in Lect21.pdf)

• If ${V}$ has a ${T}$-invariant subspace, then we are able to find a matrix representation in nice block form. (See p.455.)

After Class Exercises: Ex 9.3 Qn 2(b), 3, 5 (See L21-ace.pdf)