Sumer 2017 MATH1231 Page 2 USE A SEPARATE BOOK CLEARLY MARKED QUESTION 1 Suppose
ID: 3169354 • Letter: S
Question
Sumer 2017 MATH1231 Page 2 USE A SEPARATE BOOK CLEARLY MARKED QUESTION 1 Suppose in Ps (E), the vector space of real polynomials of degree 3 or less that 1. i) p1(x) = 1-x+z" P5(2) = 1-2z-z? and S = {n(z), P2(z), p3(2),P4(a),P5(x)) Answer the questions (a) to (d) below giving brief reasons for each part The Maple given, or your own hand calculations, can be used in parts (b), (c) and (d) with (LinearAlgebra) Id4: IdentityMatrix(4) 1 -2 -4 1 1 0 0 0 -1 32 2 01 0 0 0 1 1 0 0 0 1 0 1 3 -5 0-2 0 0 01 ReducedRowEchelonForm (AI); 1 0 -2 0 7 3 0 0 2 01 1 0 31 0 0 -1 0 0 0 1 3 1 0-1 -1 0 0 0 0 0 01 21 a) Without using the Maple above, or solving any system of equations, b) Write one of the polynoials in S as a lnea obination of the c) Find conditions on (0, al(J2: (13 such that q(a)-ao-u_T+(1212+a323 d) Find a basis for P3(R) that contains as many of the polynomials carefully explain why S cannot be a linearly independent set others. belongs to span(S) P)Ps(x) as possible Please see over . .Explanation / Answer
1(a). P3, the set of all polynomials of degree 3 or less has the standard basis {1,x,x2,x3}. The dimension of P3 is 4. Since the set S has 5 vectors(polynomials), it cannot be linearly independent.
(b).Let A =
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It may be observed that theentries in the columns of A are the scal;ar multiples of 1 and the coefficients of x,x2,x3 in the vectors(polynomials) in S.To examine whether one of the polynomials in S can be expressed as a linear combination of others, we will reduce A to its RREF. The RREf of A is
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It is now apparent that p3(x)= -2p1(x)+p2(x)+0p4(x)+0p5(x) and p5(x)= 7p1(x)+3p2(x)+0p3(x)+3p4(x).
(c). If q(x) = a0+a1x+a2x2+a3x3 span{S} , then q(x) has to be a linear combination of p1(x),p2(x),p3(x),p4(x) and p5(x) or a linear combination of p1(x),p2(x) andp4(x) as p3(x) and p5(x) are linear combinations of these polynomials.Let q(x)=a0+a1x+a2x2+a3x3=c1p1(x)+c2p2(x)+c4p4(x)=c1(1-x+x3)+c2(-2+x+x2-3x3)+c4(2x-x2)= (c1-2c2) +x(-c1+c2)+x2(c2-c4)+x3(c1-3c2) or, a0 = c1-2c2,a1= c2-c1, a2= c2-c4 and a3= c1-3c2 where c1,c2,and c4 are real numbers.
(d). From part(b) above, a basis for P3® that contains as many of the polynomials p1(x),p2(x),p3(x),p4(x) andp5(x) is { p1(x),p2(x),p4(x)}.
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