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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?

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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 $$ x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \in \mathbb{R}^2. $$

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