Verify that T is a linear transformation

Question

Define $$ T\pmatrix { \begin {bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} }=\begin {bmatrix} x_1+x_3 \\ x_2+x_3 \end{bmatrix}$$ Verify that T is a linear transformation

I know that this is not the best format, and I understand if you are frustrated, but if you can resist, please don't post a comment just telling me that I should write my questions better.

I have just a few questions: Since there are three inputs, shouldn't there be three outputs as well? Is there a zero in there somewhere? Like, should I read it like this: T(x1, x2, x3) = (x1 + x2, 0, x2 + x3), where the zero is in the middle? How is it supposed to be read?

Answer 1

A function is a linear transformation if

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

Does this satisfy those two rules?

As for your other question, why can't a function have a different codomain from its domain?

Isn't $f(x,y) = x+y$ a valid function (and a linear map).

Going the other way, $f(t) = (t, 2t, 3t)$ is also a linear map.

Answer 2

No, your function $T$ takes inputs from $\mathbb{R}^3$ and has outputs in $\mathbb{R}^2$.

Answer 3

$$T (1,2,3)=(4,5)$$

the image of a triplet is a couple.

Answer 4

Notice that $$ T\left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) = \begin{bmatrix} x_1 + x_3 \\ x_2 + x_3 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix} \cdot \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, $$ and recall that every matrix transformation is a linear transformation.