Let H= {p(t): p(t)= a+ bt+ ct°; a,b,c ER } (a) Show that H is a subspace of P3: (b) Let p 1, P2, P3 be polynomials in H, such that p,(t) = 2, p2(t)=1+3f° , p3(t)= – 1-t-8. Use coordinate vectors in each of the following andjustify your answer each part. (i) Verify that {p1, P2, P3} form a linearly independent set in P3 (ii) Verify that {p, P2, P3} does not span P3. (iii) Can the set {p,, pɔ, p3} form basis for P3? 11 (c) Let T:H→ R* be a linear transformation, defined by p(0) - p(0) T(p) = p(0) Lp(0) Find Ker T and find two polynomials that span Ker T.

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Chapter2: Second-order Linear Odes
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Let H= {p(t): p(t)= a+ bt+ ct°; a,b,c ER }
(a)
Show that H is a subspace of P3:
(b) Let p 1, P2, P3 be polynomials in H, such that
p,(t) = 2, p2(t)=1+3f° , p3(t)= – 1-t-8. Use coordinate
vectors in each of the following andjustify your answer each
part.
(i)
Verify that {p1, P2, P3} form a linearly independent
set in P3
(ii)
Verify that {p, P2, P3} does not span P3.
(iii)
Can the set {p,, pɔ, p3} form basis for P3?
11
(c)
Let T:H→ R* be a linear transformation, defined by
p(0) -
p(0)
T(p) =
p(0)
Lp(0)
Find Ker T and find two polynomials that span Ker T.
Transcribed Image Text:Let H= {p(t): p(t)= a+ bt+ ct°; a,b,c ER } (a) Show that H is a subspace of P3: (b) Let p 1, P2, P3 be polynomials in H, such that p,(t) = 2, p2(t)=1+3f° , p3(t)= – 1-t-8. Use coordinate vectors in each of the following andjustify your answer each part. (i) Verify that {p1, P2, P3} form a linearly independent set in P3 (ii) Verify that {p, P2, P3} does not span P3. (iii) Can the set {p,, pɔ, p3} form basis for P3? 11 (c) Let T:H→ R* be a linear transformation, defined by p(0) - p(0) T(p) = p(0) Lp(0) Find Ker T and find two polynomials that span Ker T.
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