In this question we denote by P2(R) the set of functions (a bx ca2: a, b, cER, w

Question

In this question we denote by P2(R) the set of functions (a bx ca2: a, b, cER, which is a vector space under the usual addition and scalar multiplication of functions. Let pi, p2, p3 E a) (4 marks) Prove that p,P2,Ps is a basis for P2(R). (You are free to assume that the b) (2 marks) Use your working from a) to write the function p(x) 8+2r-52 as a linear P2(R) be given by pi(x) 2, P2(x) 4-2, and pa(x) 32 x2, and p3(z functions 1, r, and x2 are linearly independent.) combination of pi, p2, and p3.

Explanation / Answer

a). Let A =

2

4

3

1

0

1

0

-1

1

It may be observed that the entries in the columns of A are the scalar multiples of 1 and the coefficients of x,x2 in p1(x), p2(x) and p3(x), respectively.

The RREF of A is I3which implies that p1(x), p2(x) and p3(x) are linearly independent and span P2. Therefore, { p1(x), p2(x),p3(x)} is a basis for P2.

b). Let B = [A|p(x)] =

2

4

3

8

1

0

1

2

0

-1

1

-5

The RREF of B is

1

0

0

26/5

0

1

0

9/5

0

0

1

-16/5

Thus, p(x) = (26/5)p1(x)+(9/5)p2(x)-(16/5)p3(x).

2

4

3

1

0

1

0

-1

1

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