Problem 3: Consider the set of polynomials {-1+3t – 2t“, 4 – 10t + 3t², 4 – 8t – 2t“} . Let p,(t) = – 1+ 3t – 21², p2(t) = 4 – 10t + 31², and p3(t) = 4 – 8t – 212. Let H = span {p,(t), P2(t), P3(t)}, which is a subset of P2. (a) Is the polynomial 10 – 12t – 25t2 in H? Answer this question directly without the use of the coordinate mapping. WhatI mean by directly is: can you find a, b, and, c such that ap (t) + bp2(t) +cp3(t) = 10 – 12t – 25t2. If you answer yes, write 10 – 12t – 25t2 as a linear combination of p,(t), P2(t), and, p3(t). If no, make sure your work makes clear why.
Problem 3: Consider the set of polynomials {-1+3t – 2t“, 4 – 10t + 3t², 4 – 8t – 2t“} . Let p,(t) = – 1+ 3t – 21², p2(t) = 4 – 10t + 31², and p3(t) = 4 – 8t – 212. Let H = span {p,(t), P2(t), P3(t)}, which is a subset of P2. (a) Is the polynomial 10 – 12t – 25t2 in H? Answer this question directly without the use of the coordinate mapping. WhatI mean by directly is: can you find a, b, and, c such that ap (t) + bp2(t) +cp3(t) = 10 – 12t – 25t2. If you answer yes, write 10 – 12t – 25t2 as a linear combination of p,(t), P2(t), and, p3(t). If no, make sure your work makes clear why.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 13E
Related questions
Question
![Problem 3: Consider the set of polynomials {-1+ 31 – 21?, 4 – 10t + 31², 4 – 8t – 212} .
Let p,(1) = – 1+ 3t – 21?, p2(t) = 4 – 10t + 3t², and p3(t) = 4 – 8t – 21². Let
H = span {p,(t), p2(t), p3(t)}, which is a subset of P,.
%3D
%3D
(a) Is the polynomial 10 – 12t – 25t2 in H? Answer this question directly without the use of
the coordinate mapping. What I mean by directly is: can you find a, b, and, c such that
ap (t) + bp2(t) + cp3(t) = 10 – 12t – 25t2. If you answer yes, write 10 – 12t – 25t2 as
a linear combination of p,(t), p2(t), and, p3(t). If no, make sure your work makes clear
why.
Hint: You can address this similar to partial fraction decomposition (you'll collect terms
and equate coefficients). It will lead to a system of equations which you can solve using
Math 6 techniques (matrix row reductions). Please do show all work (you can and should
use technology to row reduce any matrices).
(b) Use the standard basis for P, B = {1, t, t²}, and the coordinate mapping to show that
10 – 12t – 25t² is in H, and use this to write 10 – 12t – 25t2 as a linear combination of
P¡(t), P2(t), and, p3(t). Please show all work!
(c) Is the polynomial 3 – 3t in H? If you answer yes, write 3 – 3t as a linear combination of
P,(t), p2(t), and, p3(t). If no, make sure your work is clear as to why not.
(d) Does the set of polynomials {-1 + 3t – 2t², 4 – 10t + 3t², 4 – 8t – 21²} span P,? Why
or why not? Answer this question without using a coordinate mapping. (You can address
something you've done above to answer this...!)
(e) Is the set of polynomials {-1+ 3t – 2t², 4 – 10t + 3t², 4 – 8t – 2t2} independent?
Answer this question directly without using a coordinate map.
[Hint: Can you find scalars a,b, and c, not all zero, such that
a (-1+3t – 212) + b (4 – 10t + 31²) + c (4 – 8t – 21²) = 0 + 0t + Or².
terms will again lead you to 3x3 linear system. Fell free to reuse some work from part (a)
to save time!]
Collecting
If the answer is no (hint, it is), find a dependence relation for the polynomials. (Check
your answer).
(f) Give a basis for the subspace H of P2, where
H
= span {-1+3t – 2t², 4 – 10t + 3t², 4 – 8t – 2t²}
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Transcribed Image Text:Problem 3: Consider the set of polynomials {-1+ 31 – 21?, 4 – 10t + 31², 4 – 8t – 212} .
Let p,(1) = – 1+ 3t – 21?, p2(t) = 4 – 10t + 3t², and p3(t) = 4 – 8t – 21². Let
H = span {p,(t), p2(t), p3(t)}, which is a subset of P,.
%3D
%3D
(a) Is the polynomial 10 – 12t – 25t2 in H? Answer this question directly without the use of
the coordinate mapping. What I mean by directly is: can you find a, b, and, c such that
ap (t) + bp2(t) + cp3(t) = 10 – 12t – 25t2. If you answer yes, write 10 – 12t – 25t2 as
a linear combination of p,(t), p2(t), and, p3(t). If no, make sure your work makes clear
why.
Hint: You can address this similar to partial fraction decomposition (you'll collect terms
and equate coefficients). It will lead to a system of equations which you can solve using
Math 6 techniques (matrix row reductions). Please do show all work (you can and should
use technology to row reduce any matrices).
(b) Use the standard basis for P, B = {1, t, t²}, and the coordinate mapping to show that
10 – 12t – 25t² is in H, and use this to write 10 – 12t – 25t2 as a linear combination of
P¡(t), P2(t), and, p3(t). Please show all work!
(c) Is the polynomial 3 – 3t in H? If you answer yes, write 3 – 3t as a linear combination of
P,(t), p2(t), and, p3(t). If no, make sure your work is clear as to why not.
(d) Does the set of polynomials {-1 + 3t – 2t², 4 – 10t + 3t², 4 – 8t – 21²} span P,? Why
or why not? Answer this question without using a coordinate mapping. (You can address
something you've done above to answer this...!)
(e) Is the set of polynomials {-1+ 3t – 2t², 4 – 10t + 3t², 4 – 8t – 2t2} independent?
Answer this question directly without using a coordinate map.
[Hint: Can you find scalars a,b, and c, not all zero, such that
a (-1+3t – 212) + b (4 – 10t + 31²) + c (4 – 8t – 21²) = 0 + 0t + Or².
terms will again lead you to 3x3 linear system. Fell free to reuse some work from part (a)
to save time!]
Collecting
If the answer is no (hint, it is), find a dependence relation for the polynomials. (Check
your answer).
(f) Give a basis for the subspace H of P2, where
H
= span {-1+3t – 2t², 4 – 10t + 3t², 4 – 8t – 2t²}
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