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Applied Mathematics 1411A/B Chapter Notes - Chapter 6.2: Orthogonal Complement, Triangle Inequality

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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

6. 2 angle & orthogonality in inner product spaces. If and are vectors in a real inner product space : If , , are vectors in a real inner product space , and if is any scalar: Two vectors and in an inner product space called orthogonal if. If and are orthogonal vectors in a real inner product space: If is a subspace of a real inner product space , then the set of all vectors in that are orthogonal to every vector in is called the orthogonal complement of and is denoted by. If is a subspace of a real inner product space : is a subspace of. If is a subspace of a real nite-dimensional inner product space , then the orthogonal complement of is.

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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.

8. Determine if each of the following statements is true or false. Clearly indicate your answer for each part by stating if it is true or false no justification is required). Each part is worth 2 marks for a correct answer; -1 for an incorrect answer to discourage guessing) (a) If A and B are n x n matrices, then rank(BA) rank(A). (b) If {A, B, C} is a linearly independent set of n x m matrices, then {AT, BT,CT} is a linearly independent set of m x n matrices. (C) W = {A E Rnxn A is invertible } is a subspace of Rnxn, where Rnxn denotes the vector space of real n x n matrices. (d) If xi and X2 are orthogonal vectors in R3 then it is always possible to find a unique vector X3 such that {x1, X2, X3} is an orthogonal basis for R3. (e) Let W be a subspace of R" and let x ER". Then projw(x - projw(x)) = 0.

Please Show Work.

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