Mathematical Physics Vol 1

Chapter 1. Vector algebra

36

Definition Orthogonal basis for the vector space is a basis in which all vectors { u 1 ,..., u k } are mutually orthogonal.

Theorem2 Let { u 1 ,..., u k } be an orthogonal basis in R k . Each vector in this space can be represented as u = c 1 u 1 + ··· + c k u k , where the set { c 1 ,..., c k } is a representation of the vector u in space R k with the basis { u 1 ,..., u k } . Then, for each i ∈{ 1 ,..., k } , c i = u · u i / ( u i · u i ) is the Fourier coefficient of vector u with respect to vector u i .

Proof

u = c 1 u 1 + ··· + c k u k |· u i ⇒ u · u i = c i ( u i · u i ) ⇒ c i = u · u i ( u i · u i ) .

Theorem 3 (Gram-Schmidt orthogonalization) If { u 1 , ··· , u k } , for k ≥ 1, is the basis pf space R

k , then the vectors

p 1 = u 1 , p 2 = u 2 − proj ( u 2 , p 1 ) , .. . = .. . , p k = u k − proj ( u k , p 1 ) −···− proj ( u k , p k − 1 )

form the orthogonal basis of that space.

Proof The proof consists of two parts: - part 1 prove that vectors p k are not zero-vectors, - part 2 prove that the vectors p 1 ,..., p k are mutually orthogonal, and thus, according to Theorem 1, it follows that { p 1 ,..., p k } form an orthogonal basis of space R k . Part 1 will be proved using the principle of mathematical induction: - step one: for k = 1

Given that p 1 = u 1 , the Theorem is true. - step two: it is true for k − 1 ≥ 1, i.e. vectors p 1 ,..., p k − 1

are mutually orthogonal, non-zero vectors in a space whose

basis is { u 1 ,..., u k

.

− 1 }