Linear algebra question

Question

Let T: R³ → R³ be a linear transformation such that T(1, 0, 0) = (−1, 2, 4), T(0, 1, 0) = (−2, 1, 3),and T(0, 0, 1) = (0, 2, −2).

 

Find the indicated image. T(0, 1, −3)

Answer 1

(0, 1, -3) = 0(1, 0, 0) + 1(0, 1, 0) -3(0, 0, 1)

So, T(0, 1, -3) = 0T(1, 0, 0) +1T(0, 1, 0) -3T(0, 0, 1) = (-2, 1, 3) + (0, -6, 6) = (-2, -5, 9)

Note: Since T is a linear transformation, T(v+w) = T(v) + T(w) and T(cv) = cT(v), where c is a constant and v,w are in R³.