Exercise I.B.2

Question

Give specific linear maps $\phi,\varphi:\mathbb{R}^2	o\mathbb{R}^2$ such that $\phi\varphi 
eq \varphi\phi$.

Toğrul Topal · Accepted Answer

$
ewcommand{\Hom}[2]{\mathrm{Hom}\left( #1, #2 ight)}$
Let $\phi, \varphi$ be defined by
    $$\phi\left(\begin{bmatrix} a \ b \end{bmatrix} ight) := \begin{bmatrix} a+b \ b \end{bmatrix} \qquad  \varphi\left(\begin{bmatrix} a \ b \end{bmatrix} ight) := \begin{bmatrix} a+b \ 2b \end{bmatrix}$$
    It is easy to verify that both $\phi, \varphi \in \Hom{\mathbb{R}^2}{\mathbb{R}^2}$. But we have
    $$\phi\left(\varphi\begin{bmatrix} 1 \ 1 \end{bmatrix} ight) := \phi\begin{bmatrix} 2 \ 2 \end{bmatrix} = \begin{bmatrix} 4 \ 2 \end{bmatrix} $$
    whereas
    $$\varphi\left(\phi\begin{bmatrix} 1 \ 1 \end{bmatrix} ight) := \varphi\begin{bmatrix} 2 \ 1 \end{bmatrix} = \begin{bmatrix} 3 \ 2 \end{bmatrix} $$

Exercise I.B.2

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Exercise I.B.2

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