Problem 3: Consider the set of polynomials {-1+3t – 21², 4 – 10t + 3t², 4 – 8t – 21²} . Let p¡(t) = – 1 + 3t – 21², p2(t) = 4 – 10t + 3ť², 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+3t – 21², 4 – 10t + 3t², 4 – 8t – 21²} . Let p¡(t) = – 1 + 3t – 21², p2(t) = 4 – 10t + 3ť², and p3(t) = 4 – 8t – 21². Let H = span {p,(t), P2(t), P3(t)}, which is a subset of P,.


Extracted text: Is the set of polynomials {-1+ 3t – 21², 4 – 10t + 31², 4 – 8t – 212} 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 – 2t²) +b (4 – 101 + 3t²) + c (4 – 8t – 2r²) = 0 + 0t + Or². Collecting terms will again lead you to 3x3 linear system. Fell free to reuse some work from part (a) to save time!] If the answer is no (hint, it is), find a dependence relation for the polynomials. (Check your answer). Give a basis for the subspace H of P, where 21 H = span {-1+ 3t – 2t2, 4 – 10t + 3t?, 4 – 8t – 212}