Applied Mathematics 1411A/B Chapter Notes - Chapter 6.3: Coordinate Vector, Unit Vector

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scarlethound445

3 Dec 2018

School

Western University

Department

Applied Mathematics

Course

Applied Mathematics 1411A/B

Professor

Bogdan Tudose

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Document Summary

2 vectors in an inner product space are orthogonal if their inner product is 0. A set of 2 or more vectors in a real inner product space is orthogonal if all pairs of distinct vectors in the set are orthogonal norm 1 (unit vector) is linearly independent. An orthogonal set in which each vector has norm 1 is said to be orthonormal. Normalizing: to convert an orthogonal set of nonzero vectors into an orthonormal set, Coordinates relative to orthonormal bases multiply each vector in the orthognal set by the reciprocal of its length to create a vector of. If is an orothognal set of nonzero vectors in an inner product space, then. If is an orthogonal basis for an inner product space , and if is any vector in , then. If is an orthonromal basis for an inner product space , and if is any vector in , then.

Related Questions

1. Let V be an inner product space. Prove the Pythagoreantheorem: if v and w E V are orthogonal then ||v+w||^2 =||v||^2 + ||w||^2. (Suggestion: start with ||v+w||^2 = (v+w,v+w).)

2. Let V=P2, the polynomials of degree less than/equal to 2 withcoefficients in R, and let (.,.): VxV -> R be the map

(p,q) = p(0)q(0) + p(1)q(1) + p(2)q(2) for any two polynomialsp(x),q(x) E V. So, for example,

(x+2, x^2-1) = (0+2)(0^2-1) + (1+2)(1^2 - 1) + (2+2)(2^2-1) =10.

a) Show that the function(.,.) defines an inner product onV.

b) Is the set A=(1,x,x^2) an orthogonal set? (with respect tothe inner product above.)

c) Is the set B=(1,x-1,3x^2-6x+1) an orthogonal set?

d) Calculate ||1||^2, and ||3x^2-6x+1||^2

e) Is the set B an orthonormal set?

f) Write f=x^2-6x+12 as a linear combination of the vectors inB.

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Applied Mathematics 1411A/B Chapter Notes - Chapter 6.3: Coordinate Vector, Unit Vector

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