Determine whether transformation is linear and finding its matrix?

Question

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?

Answer 1

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.