Question

Determine whether the function T: R

3 →R

3 by 𝑇 ([

𝑥

𝑦

𝑧

]) = [

𝑥 + 𝑦 − 𝑧

2𝑥𝑦

𝑥 + 𝑧 + 1

] is a linear 

transformation between vector space.

Answer 1

The given function-

$T(\begin{bmatrix} x\\y\\z \end{bmatrix})=\begin{bmatrix}x+y-z\\2xy\\x+z+1\end{bmatrix}$

    Let$\alpha=\begin{bmatrix} \alpha_1\\\alpha_2\\\alpha_3 \end{bmatrix},\beta=\begin{bmatrix} \beta_1\\\beta_2\\\beta_3 \end{bmatrix}\in R^3$

$T(\alpha+\beta)=T(\begin{bmatrix} \alpha_1\\\alpha_2\\\alpha_3 \end{bmatrix}+\begin{bmatrix} \beta_1\\\beta_2\\\beta_3 \end{bmatrix}) \\[9pt] =T(\begin{bmatrix} \alpha_1+\beta_1\\\alpha_2+\beta_2\\\alpha_3+\beta_3 \end{bmatrix}) \\[9pt] =\begin{bmatrix} \alpha_1+\beta_1+\alpha_2+\beta_2-(\alpha_3+\beta_3)\\2(\alpha_1+\beta_1)(\alpha_2+\beta_2)\\\alpha_1+\beta_1+\alpha_3+\beta_3+1 \end{bmatrix}\\[9pt] =\begin{bmatrix} \alpha_1+\alpha_2-\alpha_3\\2\alpha_1 \alpha_2\\\alpha_1+\alpha_3+1 \end{bmatrix}+\begin{bmatrix} \beta_1+\beta_2-\beta_3\\2\beta_1 \beta_2+2(\alpha_1 \beta_2+\beta_1 \alpha_2)\\\beta_1+\beta_3 \end{bmatrix}\\[9pt]\neq T(\alpha)+T(\beta)$

Since, $T(\beta)=\begin{bmatrix} \alpha_1+\alpha_2-\alpha_3\\2\alpha_1 alpha_2\\\alpha_1+alpha_3+1 \end{bmatrix}\neq\begin{bmatrix} \beta_1+\beta_2-\beta_3\\2\beta_1 \beta_2+2(\alpha_1 \beta_2+\beta_1 \alpha_2)\\\beta_1+\beta_3 \end{bmatrix}$

Therefore T is not liner transformation between vector space.