Kunze Linear Algebra · Problem 8.3.6

Inner Product Spaces

Mathematics — Methods · adjoints-rank-one-operators

Problem

Let

V

be an inner product space and

\beta

\gamma

fixed vectors in

V

. Show that

T\alpha = (\alpha|\beta)\gamma

defines a linear operator on

V

. Show that

T

has an adjoint, and describe

T^*

explicitly. Now suppose

V

C^n

with the standard inner product,

\beta = (y_1, \ldots, y_n)

, and

\gamma = (x_1, \ldots, x_n)

. What is the

j, k

entry of the matrix of

T

in the standard ordered basis? What is the rank of this matrix?