# How to determine whether transformation is linear and finding its matrix?

Consider the transformation T from $\mathbb{R}^2$ to $\mathbb{R}^3$ defined by $$T[x_1, x_2]=x_1[1,2,3]+x_2[4,5,6].$$

Is the transformation linear? If so, find its matrix.

How would I determine whether the transformation is linear and then go about obtaining its matrix?

If $A$ is an $m$-by-$n$ matrix with columns $a_1,\dots,a_n$ and $x \in \mathbb{R}^n$, then $$Ax := \sum_{i=1}^n x_i a_i.$$
Thus, the transformation $T$ above is, by definition, a matrix transformation (and, hence, a linear transformation) and $$T(x) = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix} x,$$ with $$x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \in \mathbb{R}^2.$$