# an cosntl 311 2 to obtain this expression

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: element f of V is of unit length if f , f = 1 and that two elements f and g of V are perpendicular if f , g = 0. In this terminology, the formulae (3.4), (3.5), and (3.6) can be expressed by stating that the functions √ 1/ 2, cos t, cos 2t, cos 3t, . . . , sin t, sin 2t, sin 3t, . . . are of unit length and perpendicular to each other. Moreover, by the theorem in the preceding section, any element of of V can be written as a (possibly inﬁnite) superposition of these functions. We will say that the functions 1 e0 (t) = √ , 2 e1 (t) = cos t, e1 (t) = sin t, ˆ e2 (t) = cos 2t, e2 (t) = sin 2t, ˆ ... , ... make up an orthonormal basis for V . We saw in Section 2.3 that if b1 , b2 , . . . , bn is an orthonormal basis for R n , then f ∈ R n ⇒ f = (f · b1 )b1 + · · · + (f · bn )bn . 69 The same formula holds when R n is replaced by V and the dot product is replaced by the inner product ·, · : If f is any element of V , we can write f (t) = f (t), e0 (t) e0 (t) + f (t), e1 (t) e1 (t) + f (t), e2 (t) e2 (t) + · · · ˆ ˆ ˆ + f (t), e1 (t)...
View Full Document

## This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online