# 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?

• Use the definition or some properties/results/lemmata/Theorems about linear maps. – user251257 Feb 8 '18 at 13:34
• "Find its matrix" depends on the bases chosen for the domain and target spaces. If the basis on the domain is $(1,0)^T$ and $(0,1)^T$ and the one on the target $(1,0,0)^T$, $(0,1,0)^T$, $(0,0,1)^T$, then the matrix would be formed by putting $[1,2,3]$ and $[4,5,6]$ as columns, in that order. – user525761 Feb 8 '18 at 13:37

• $T(v+u) = T(v) + T(u)$
• $T(\alpha u) = \alpha T(u)$