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]^β_α.

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]^α_α and complete the following formula
(UT)(a+bx+cx²) =

Expert Solution

Here we use the theory of verifying a basis or not. Matrix representation of linear transformation and coordinate basis representation etc. The last part provides some incomplete data.

Colby Messinger answered 1 year ago

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