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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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    $\begingroup$ Use the definition or some properties/results/lemmata/Theorems about linear maps. $\endgroup$ – user251257 Feb 8 '18 at 13:34
  • $\begingroup$ "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. $\endgroup$ – user525761 Feb 8 '18 at 13:37
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Just check using the definition of a linear transformation:

  • $T(v+u) = T(v) + T(u)$

  • $T(\alpha u) = \alpha T(u)$

The matrix of a transformation depends on the basis vectors you choose to use, but the simplest case would be to use the standard basis - (1, 0) and (0, 1).

Now ask yourself this - if I want the transformation matrix to send (1, 0) to (1, 2, 3) and (0, 1) to (4, 5, 6), what should the matrix look like? Use the definition of matrix multiplication. Hint - it'll be a 3 by 2 matrix.

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