Question

Let T: P?(?) ? ?3 be given by f ? ? f(1) ? ? f(2) ?. ? f(3) ? 1. Find the matrix [T]?? when ? = {1, x, x²} and ? = ? ? 1 ?, ? 0 ?, ? 0 ? ?. ? ? 0 ? ? 1 ? ? 0 ? ? ? ? 0 ? ? 0 ? ? 1 ? ? 2. Find the matrix [T]?? when ? = {1, x, x²} and ? = ? ? 1 ?, ? 1 ?, ? 1 ? ?. ? ? 1 ? ? 2 ? ? 4 ? ? ? ? 1 ? ? 3 ? ? 9 ? ? 3. Prove that T is one-to-one. 4. Find a basis ? for P?(?) such that if ? is the standard basis for ?3, then [T]?? = ? 1 0 0 ? ? 0 1 0 ?. ? 0 0 1 ?

          Let T: P?(?) ? ?3 be given by f ? ? f(1) ?
? f(2) ?.
? f(3) ?

1. Find the matrix [T]?? when ? = {1, x, x²} and ? = ? ? 1 ?, ? 0 ?, ? 0 ? ?.
? ? 0 ? ? 1 ? ? 0 ? ?
? ? 0 ? ? 0 ? ? 1 ? ?

2. Find the matrix [T]?? when ? = {1, x, x²} and ? = ? ? 1 ?, ? 1 ?, ? 1 ? ?.
? ? 1 ? ? 2 ? ? 4 ? ?
? ? 1 ? ? 3 ? ? 9 ? ?

3. Prove that T is one-to-one.
4. Find a basis ? for P?(?) such that if ? is the standard basis for ?3, then
[T]?? = ? 1 0 0 ?
? 0 1 0 ?.
? 0 0 1 ?

Let T: P?(?) ? ?3 be given by f ? ? f(1) ?
? f(2) ?.
? f(3) ?

1. Find the matrix [T]?? when ? = 1, x, x² and ? = ? ? 1 ?, ? 0 ?, ? 0 ? ?.
? ? 0 ? ? 1 ? ? 0 ? ?
? ? 0 ? ? 0 ? ? 1 ? ?

2. Find the matrix [T]?? when ? = 1, x, x² and ? = ? ? 1 ?, ? 1 ?, ? 1 ? ?.
? ? 1 ? ? 2 ? ? 4 ? ?
? ? 1 ? ? 3 ? ? 9 ? ?

3. Prove that T is one-to-one.
4. Find a basis ? for P?(?) such that if ? is the standard basis for ?3, then
[T]?? = ? 1 0 0 ?
? 0 1 0 ?.
? 0 0 1 ?

Added by Claudia M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

08/20/2023

Step 1

Find the matrix [T]B when B = {1, x, x^2}. To find the matrix representation of T with respect to the basis B, we need to find the images of the basis vectors under T and express them as linear combinations of the basis vectors. T(1) = f(1) + f(2) + f(3) = 1 + 2 Show more…

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Let T: P2(R) → R3 be given by f ℒ (f(1), f(2), f(3)). 1. Find the matrix [T]̧̑ when ̑ = {1, x, x2} and ̑ = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}. 2. Find the matrix [T]̧̑ when ̑ = {1, x, x2} and ̑ = {(1, 1, 1), (1, 2, 3), (1, 4, 9)}. 3. Prove that T is one-to-one. 4. Find a basis ̑ for P2(R) such that if ̑ is the standard basis for R3, then [T]̧̑ = I.