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Exercise 3

Avg difficulty 3.0

Consider this transformation of C3\mathbb{C}^{3}

t((xyz))=(x−zz2y)t(\begin{pmatrix}x\\ y\\ z\end{pmatrix})=\begin{pmatrix}x-z\\ z\\ 2y\end{pmatrix}

and these bases.

B=⟨(123),(010),(001)⟩D=⟨(100),(110),(101)⟩B=\langle\begin{pmatrix}1\\ 2\\ 3\end{pmatrix},\begin{pmatrix}0\\ 1\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}\rangle\qquad D=\langle\begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}1\\ 1\\ 0\end{pmatrix},\begin{pmatrix}1\\ 0\\ 1\end{pmatrix}\rangle

We will compute the parts of the arrow diagram to represent the transformation using two similar matrices.

  • Draw the arrow diagram, specialized for this case.

  • Compute T=RepB,B(t)T={\rm Rep}_{B,B}(t).

  • Compute T^=RepD,D(t)\hat{T}={\rm Rep}_{D,D}(t).

  • Compute the matrices for the other two sides of the arrow square.

Hefferon, Linear Algebra, Similarity, Definition and Examples, Exercise 3

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Adapted from Linear Algebra © 2016 Jim Hefferon. Licensed under CC BY-SA 3.0 US. Source: Linear Algebra. Imported at revision df2262e.