MATH115 Lecture Notes - Lecture 28: Linear Combination, Orthogonal Matrix

33 views4 pages

scarletscorpion275

10 Oct 2015

School

University of Waterloo

Department

Mathematics

Course

MATH115

Professor

Tina Davidson

For unlimited access to Class Notes, a Class+ subscription is required.

Document Summary

Friday, november 7 lecture 28 : orthogonal and orthonormal basis. Expectations: define orthogonal set and orthonormal set, normalize an orthogonal set, recognize that orthogonal sets are always linearly independent, define orthogonal basis and orthonormal basis. u v = 0. 28. 1 definition two vectors u and v in n are said to be orthogonal if. 28. 2 definition a finite subset s of vectors in n is said to be an orthogonal set if x y. {v1, v2, v3} = { (1, 0, 1), (0, 1, 0), ( 2, 0, 2)} is an orthogonal set but not orthonormal. To do this show that the dot-product any pair of vectors and see that it equals 0. The norm of (1, 0, 1) is root 2 and so the set is not orthonormal. ) 28. 2. 2 remark given any orthogonal set we can obtain from it an orthonormal set by. That is given any vector u, (1/||u||) u has norm 1.

Related Questions

Show transcribed image text

orangepig166

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.