a 2D change of basis: a non-orthonormal basis

suppose we work a slightly different problem. suppose, as before, that we have new basis vectors rotated 30{}^{\circ} 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

(\sqrt{3},\ -1)

(\frac{1}{4},\ \frac{\sqrt{3}}{4})

here’s what the new basis looks like wrt the old one:


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