Kunze Linear Algebra · Problem 8.2.6

Inner Product Spaces

Mathematics — Methods · orthogonal-projection

Problem

Let

W

be the subspace of

R^2

spanned by the vector

(3, 4)

. Using the standard inner product, let

E

be the orthogonal projection of

R^2

onto

W

. Find (a) a formula for

E(x_1, x_2)

; (b) the matrix of

E

in the standard ordered basis; (c)

W^\perp

; (d) an orthonormal basis in which

E

is represented by the matrix

\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}.