Question

Determine if the given function is a linear transformation. t(x1, x2, x3) = (−2x2, 6x3)

Answer 1

$T:\mathbb R^3\to\mathbb R^2$ will be linear if for any $\mathbf x,\mathbf y\in\mathbb R^3$ and constants $a,b\in\mathbb R$, we have

$T(a\mathbf x+b\mathbf y)=aT(\mathbf x)+bT(\mathbf y)$

Let $\mathbf x=(x_1,x_2,x_3)$ and $\mathbf y=(y_1,y_2,y_3)$, and let $a,b$ be any two scalars. Then

$a\mathbf x+b\mathbf y=(ax_1+by_1,ax_2+by_2,ax_3+by_3)$

By definition of $T$, we have

$T(a\mathbf x+b\mathbf y)=(-2ax_2-2by_2,6ax_3+6by_3)$
$T(a\mathbf x+b\mathbf y)=(-2ax_2,6ax_3)+(-2by_2,6by_3)$
$T(a\mathbf x+b\mathbf y)=a(-2x_2,6x_3)+b(-2y_2,6y_3)$
$T(a\mathbf x+b\mathbf y)=aT(\mathbf x)+bT(\mathbf y)$

Therefore $T$ is a linear transformation.