MATH307 December 2012
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q3 (c) • Q3 (d) • Q3 (e) • Q4 (a) • Q4 (b) • Q4 (c) • Q4 (d) • Q4 (e) • Q5 (a) • Q5 (b) • Q5 (c) • Q5 (d) • Q6 (a) • Q6 (b) • Q6 (c) • Q6 (d) • Q6 (e) • Q6 (f) •
Question 01 (d)
In this question we will work with polynomials of degree 3 written
(d) Using the equation in (c) find the equation Cs=c satisfied by if p(x)
(i) has coefficient vector (and therefore has zero slope at x = 0 and x = 2).
(ii) passes through the points (0,1), (1,2) and (2,2).
Does this equation have a solution? Give a reason.
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Using the equation from part (c), we have
Notice that for there to be a solution we need in the range of C, R(C). We know that R(C) is the orthogonal complement to the nullspace of , . Therefore, onsider the basis vector of
In other words, is not orthogonal to . This means that cannot possibly be in the range of C and so there will be no solution to the problem.
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