Here is how I did it: I built a basis of $u, v, w$ $u = 1*u + 0*v + 0*w$ which gives me (1, 0, 0) $v = 0*u + 2*v + 0*w$ which gives me (0,2,0) $w = 0*u + 0*v - $ which gives me $(0, 0, -1)
I then made matrix of them in basis B and turned it to standard basis through $A_s = PA_BP^-1$ where P is made of the given vectors and got $\begin{bmatrix} -7 \, 8 \, 10 \\ -4 \, -5 \, -8 \\ -1 \, -1\ , 0 \end{bmatrix}$ This is correct but I have a question about the transformation matrix in basis B, because it doesn't give me $T(u) = Au = u$ or any of the other requirements? Why?
Also my answer sheet did it in a much easier way.
Here is how they did
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Why did they just multiply each vector by its eigenvalue and create the B matrix? I don't see how that could be the matrix of basis B and why is it different from mine? Did I just get lucky when I got the right answer?