Question

Find the transition matrix corresponding to the change of basis for [u1, u2] to [e1, e2].

u1= (1,2)^T u2 = (2, 5)^T

Answer 1

i think this is another way to say "find the inverse of
\[\begin{pmatrix}
1 & 2\\
2 & 5
\end{pmatrix}
\]
and since the determinant here is 1, answer is
\[\begin{pmatrix}
5&-2\\
-2 &1
\end{pmatrix}
\]