suppose we work a slightly different problem. suppose, as before, that we have new basis vectors rotated 
CW from the old ones, but suppose the new x basis vector is of length 2, and the new y basis vector is of length 1/2. that is, the new basis vectors are given as
CW from the old ones, but suppose the new x basis vector is of length 2, and the new y basis vector is of length 1/2. that is, the new basis vectors are given as
here’s what the new basis looks like wrt the old one:
Rip's Applied Mathematics Blog 