Let T : R3 → R3 be the linear operator which rotates each vector about the y-axis by an angle 90 . Find two non-zero T -invariant subspaces U1 and U2 such that R3 = U1 ⊕ U2 (prove this). Find a basis S of R3 such that S is different from the standard basis E and the matrix of T in basis S is block diagonal. Find the matrix of T in basis S.
So far I have got the transformation T(x,y,z) = (xcosθ + zsinθ, y , zcosθ - xsinθ) where θ is pi/2.
Let u1 = (a,0,b) in the x-z plane. U1 is the x-z plane. u1 ∃ U1. T(u1) = (acosθ + bsinθ, 0 , bcosθ - asinθ) and it still in U1 under the mapping of T.
Let u2 = (0,c,0) then T(u2) = u2, then U2 = span{u2} and is T invariant.
I'm stuck from here onwards. I think I'm missing a third vector.