Mathematical Physics Vol 1

Chapter 1. Vector algebra

38

Theorem4 If the set { u 1 ,..., u k } is an orthonormalized basis, then the norm of the vector v = c 1 u 1 + ··· + c k u k , is given by the expression ∥ v ∥ = q c 2 1 + ··· + c 2 k .

Proof

∥ v ∥ = p ( c 1 u 1 + ··· + c k u k ) · ( c 1 u 1 + ··· + c k u k )= q c 2

1 + ··· + c 2 k