MATH 415 Study Guide - Final Guide: Altgeld Hall, Unit Vector, Coordinate Vector

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cerisehamster618

6 Jan 2021

School

University of Illinois

Department

Mathematics

Course

MATH 415

Professor

J Li

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Document Summary

Computations. a b c d: let v = 1 is a basis for v . (if you ignore the rst entry, it is easy to see that. : a = b + 3c 2d. 1 a b c d these vectors are linearly independent). The vector is orthogonal to v i it is orthogonal to each vector in the basis of v from a b c d which we may rewrite as. = 0 a + b = 3a + c = 2a + d = 0 and so b = a, c = 3a, d = 2a. We may conclude from this that the set of vectors orthogonal to v is precisely span. 2 and any scalar multiple of it, will be orthogonal to v . in particular, the vector. 2 (c) no, we showed in part (b) that any two vectors orthogonal to v must be scalar multiples of each other and therefore cannot be linearly independent.

Related Questions

Find the following

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.

Please Show Work.

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MATH 415 Study Guide - Final Guide: Altgeld Hall, Unit Vector, Coordinate Vector

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