16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.4.5

Inner Product Spaces

Mathematics — Linear Algebra · unitary-operators-rotation-3d

Problem

Let

V

R^3

, with the standard inner product. Let

W

be the plane spanned by

\alpha = (1, 1, 1)

and

\beta = (1, 1, -2)

. Let

U

be the linear operator defined, geometrically, as follows:

U

is rotation through the angle

\theta

, about the straight line through the origin which is orthogonal to

W

. There are actually two such rotations---choose one. Find the matrix of

U

in the standard ordered basis. (Here is one way you might proceed. Find

\alpha_1

and

\alpha_2

which form an orthonormal basis for

W

. Let

\alpha_3

be a vector of norm 1 which is orthogonal to

W

. Find the matrix of

U

in the basis

\{\alpha_1, \alpha_2, \alpha_3\}

. Perform a change of basis.)