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)




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