Question

Consider the vector space P2 := P2(F) and its standard basis α = {1,x,x^2}. 1Prove that...

Consider the vector space P2 := P2(F) and its standard basis α = {1,x,x^2}.

1Prove that β = {x−1,x^2 −x,x^2 + x} is also a basis of P2

2Given the map T : P2 → P2 defined by T(a + bx + cx2) = (a + b + c) + (a + 2b + c)x + (b + c)x2
compute [T]βα.

3 Is T invertible? Why

4 Suppose the linear map U : P2 → P2 has the matrix representation

(1 0 0

0 2 0

0 0 4)

Compute [UT]αα and complete the following formula
(UT)(a+bx+cx2) =

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Homework Answers

Answer #1

1)

Implies

So this set of 3 vectors is linearly independent and so it must be a basis

2)

and

So that

c) Yes, it is invertible as

d) So that

So that

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