C HAPTER 4 Inner Product & Orthogonality

C HAPTER O UTLINE Introduction Norm of the Vector, Examples of Inner Product Space - Euclidean n-space - Function Space, Polynomial Space Angle between Vectors Orthogonal & Orthonormal Set Normalizing Vector Gram Schmidt Process

I NNER P RODUCT S PACE Is a vector space, with an inner product Satisfy the following 3 axioms, for all vectors 1) Conjugate symmetry: 2) Linearity in the first argument: 3) Positive definiteness: with equality only for x=0.

N ORM OF THE V ECTOR ( ) Norm of the vector or length: The norm of the vector in, denoted by ; Eg: If

E UCLIDEAN N - SPACE ( ) Consider the vector space. The dot product or scalar product in is defined by: This function defines an inner product on. Eg: Find the Euclidean inner product a)

F UNCTION S PACE P OLYNOMIAL S PACE The notation - used to denote the vector space of all continuous functions on the closed interval, where. Let are functions in, an inner product on : Eg: Consider in the polynomial space with inner product

a) Find b) Find

A NGLE BETWEEN V ECTORS For any nonzero vectors u and v in an inner product space, V, the angle between u and v is defined to be the angle θ such that and Eg: Consider the vector Find the angle θ between.

O RTHOGONAL S ET Let V be an inner product space. The vectors is said to be orthogonal if Eg: Determine whether the given vectors are orthogonal

O RTHONORMAL S ET The set is said to be orthonormal if it is orthogonal and each of its vectors has norm 1, that is for all i. Eg: Let Determine whether S is an orthonormal set.

N ORMALIZING V ECTOR If or equivalently, then u is called a unit vector and is said to be normalized. To obtain a unit vector, every nonzero vector v in V, can be multiplied by the reciprocal of its length Eg: Let Normalize u and v.

G RAM S CHMIDT P ROCESS

Eg: Use Gram Schmidt Process to find an orthonormal set from the set