MATH125 Lecture Notes - Lecture 37: Orthogonal Complement, Orthogonal Matrix

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carminelocust146

20 Apr 2017

School

University of Alberta

Department

Mathematics

Course

MATH125

Professor

Nikita

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MATH125 Full Course Notes

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Document Summary

Proof. (a) q 1 = qt is orthogonal. (b) 1 = det i = det(qt q) = det qt det q = (det q)2, so that det q = 1. (c) if v is an eigenvector with eigenvalue , then ||v|| = ||qv|| = || v|| = | |||v|| so that | | = 1. (d) (q1q2)t q1q2 = qt. Rn is orthogonal to w , if v is orthogonal to every vector in w . The set of all vectors that are orthogonal to w is called the orthogonal complement of w , denoted w . W = {v in rn : v w = 0 for all w in w }. Let w be a subspace of rn: w is a subspace of rn, (w ) = w , w w = {0}, if w is spanned by vectors w1, .

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.