Kunze Linear Algebra · Problem 2.4.5

Vector Spaces

Mathematics — Methods · orthonormal-basis-coordinates

Problem

Let

\alpha = (x_1, x_2)

and

\beta = (y_1, y_2)

be vectors in

R^2

such that

x_1 y_1 + x_2 y_2 = 0, \quad x_1^2 + x_2^2 = y_1^2 + y_2^2 = 1.

Prove that

\mathfrak{B} = \{\alpha, \beta\}

is a basis for

R^2

. Find the coordinates of the vector

(a, b)

in the ordered basis

\mathfrak{B} = \{\alpha, \beta\}

. (The conditions on

\alpha

and

\beta

say, geometrically, that

\alpha

and

\beta

are perpendicular and each has length 1.)