Exercise I.B.5

Question

Let $\phi, \varphi, 	heta$ be functions given by
$$\phi: \mathbb{R}^2 	o \mathbb{R}^2, \begin{bmatrix} x \ y \end{bmatrix} \mapsto \begin{bmatrix} x \ x+y \end{bmatrix} \qquad
\varphi: \mathbb{R}^3 	o \mathbb{R}, \begin{bmatrix} x \ y \ z\end{bmatrix} \mapsto z \qquad
\phi: \mathbb{R}^2 	o \mathbb{R}, \begin{bmatrix} x \ y \end{bmatrix} \mapsto xy$$
Show that $\phi$ and $\varphi$ are linear transformations, whereas $	heta$ is not.

Toğrul Topal · Accepted Answer

\begin{enumerate}
    \item Let $[x_1,x_2], [y_1,y_2] \in \mathbb{R}^2$ and $c\in\mathbb{R}$ be arbitrary. We then have
      $$\phi \left(c \begin{bmatrix} x_1 \ x_2 \end{bmatrix} + \begin{bmatrix} y_1 \ y_2 \end{bmatrix} ight)
      = \phi \left(\begin{bmatrix} cx_1+y_1 \ cx_2+y_2 \end{bmatrix} ight) = \begin{bmatrix} cx_1 + y_1 \ cx_1 + y_1 + cx_2 + y_2 \end{bmatrix}
      = c \begin{bmatrix} x_1\ x_1 + x_2 \end{bmatrix} + \begin{bmatrix} y_1\ y_1 + y_2 \end{bmatrix}
      =  c \phi \left(\begin{bmatrix} x_1 \ x_2 \end{bmatrix}ight)  + \phi\left(\begin{bmatrix} y_1 \ y_2 \end{bmatrix} ight)$$
    \item Similarly, we have
      $$\varphi \left(c \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} + \begin{bmatrix} y_1 \ y_2 \ y_3 \end{bmatrix} ight)
      = \varphi \left(\begin{bmatrix} cx_1+y_1 \ cx_2+y_2 \cx_3 + y_3 \end{bmatrix} ight)
      = cx_3 + y_3 
      =  c \varphi \left(\begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}ight)  + \varphi\left(\begin{bmatrix} y_1 \ y_2 \ y_3 \end{bmatrix} ight)$$
    \item We have
      $$	heta \left(c \begin{bmatrix} x_1 \ x_2 \end{bmatrix} + \begin{bmatrix} y_1 \ y_2 \end{bmatrix} ight)
      = 	heta \left(\begin{bmatrix} cx_1+y_1 \ cx_2+y_2 \end{bmatrix} ight)
      = (cx_1 + y_1) (cx_2 + y_2) = c^2x_1x_2 + cx_1y_2 + cx_2y_1 + y_1y_2$$
      whereas
      $$c	heta \left(\begin{bmatrix} x_1 \ x_2 \end{bmatrix}ight) + 	heta \left(\begin{bmatrix} y_1 \ y_2 \end{bmatrix} ight)
      = cx_1x_2 + y_1y_2$$
      Setting $x_1 := 0$ and $c,x_2, y_1,y_2 := 1$ we get see the inequality of the two quantities.
    \end{enumerate}

Exercise I.B.5

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Exercise I.B.5

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