Math 417 homework 2 solutions (7pages)

(Given only for problems that are not straightforward computation; contact the instructor if you still have questions about the others.) Section 2.1 problem 1 A transformation T : R 3 → R 3 is linear only if it satisfies T(~0) = ~0. The transformation is not linear. For x1 = x2 = x3 = 0 we get y2 = 2, not y0 = 0. However, the transformation can be written T(~x) =   0 2 0 0 1 0 0 2 0     x1 x2 x3   +   0 2 0  . Outside of algebra, adding a constant vector still counts as “linear”. Section 2.1 problem 1 A linear transformation T : R 3 → R 3 must satisfy T(α · ~x) = α · ~x for all α ∈ R. But here 2T(   1 0 1  ) = 2   −1 1 1   =   −2 2 2   which is not the same as T(2   1 0 1  ) = T(   2 0 2  ) =   −2 4 2  . Note: alternatively one could check that another property of linear transformations, T(~x + ~y) = T(~x) + T(~y),