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.