MATH 404 Midterm: MATH404_HERB-R_SPRING2004_0101_MID_EXAM

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turquoisegnu128

10 Jan 2019

School

University of Maryland

Department

Mathematics

Course

MATH 404

Professor

All

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Document Summary

Math 404 - exam 1 - march 5, 2004. Let v = r[x] denote the real vector space of all polynomials with real coe cients. Determine whether or not each of the following subsets is a subspace. Justify your answers completely. (a) the set s of all f (x) v such that f (0) is an integer. (b) the set t of all f (x) v such that f (3) = 0. Let f be a eld and let v = {f (x) f [x] : deg f (x) 3}. Explain why a is algebraic over q of degree 20. De ne f (a), the sub eld obtained by adjoining a to f . (c) let a k. fix f, 6= 0, and de ne b = a. Show that f (a) = f (b). (d) let a k and b = a be as in (c).

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.