Let u1 = [1 1]^T and u2 = [0 -1]^T. Find a 2 x 2 real matrix A so that the function T_A is a map from ℝ^2 to ℝ^2, given by multiplication by A,
T_A := Av,
sends T_A(u1) = v1 and T_A(u2) = v2 where v1 = [cosθ sinθ]^T and v2 = [-sinθ cosθ]^T. Explain/justify your work.
The Attempt at a Solution
in the first part of the question, we are asked to prove that v1 and v2 form an orthonormal basis for ℝ^2. At first I thought that we would use Gram-Schmidt process but the two vectors are already orthonormal. So basically I am clueless as to where to start/proceed.