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12 Nov 2019

Suppose V is an inner product and P, Q are orthogonal projection. Prove that trace (PQ)>=0.(Linear algebra done right third edition 10.A.12)

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Patrina Schowalter
Patrina SchowalterLv2
1 Jan 2019

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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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