Question

(a) Find a linear transformation T : R2→R2 that (i) maps the x1-axis to itself, (ii) maps the x2-axis to itself, and (iii) maps no other line through the origin to itself.

For example, the negating function (n: R2→R2 defined by n(x) =−x) satisfies (i) and (ii), but not (iii).

(b) The function that maps (x1, x2) to the perimeter of a rectangle with side lengths x1 and x2 is not a linear function. Why?

For part (b) I can't come up with any counterexamples that show T(x+y) = T(x) + T(y) or that aT(x) = T(ax) isn't true, and when I tried to use a variables instead of numbers, I ended up showing that it did satisfy both conditions. I'm not sure what I'm missing.

Answer 1

First we take x₁ as X and x₂ as Y. Then