Let u be some nonzero vector in Rn L Spanu and Tx projLx P

Let u be some nonzero vector in R^n, L = Span{u}. and T(x) = proj_L(x). Prove that T is a linear transformation. For any z epsilon L, what is T(z)? Explain. If A is the standard matrix for T. what does the result of part (b) tell us about the eigenvalues, eigenvectors, and/or eigenspaces of A?

Solution

(a) For any x in Rⁿ,

                                       T(x) = <x,u>u,

where <,> denotes the inner product.

T(x+y) = <x,u> u+ <y,u> u

            =<x+y,u>u (bilinearity of <,>)

          = T(x) + T(y)

T(cx) = <cx,u>u

         =c<x,u>u (bilinearity)

         = c(Tx)

Thus T is a linear map.

(b) Clearly , if x is in L^(perp),T(x) = <x,u>u =0.

(c) As T is a projection , T² =T and its eigenvalues are 1 and 0 (the corresponding eigenspaces are span{u} and L^(perp)