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)},...

Extracted 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} | |