[/b]1. The problem statement, all variables and given/known data[/b](adsbygoogle = window.adsbygoogle || []).push({});

Let P_2 be the set of all real polynomials of degree no greater than 2.

Show that both B:={1, t, t^2} and B':= {1, 1-t, 1-t-t^2} are bases for P_2.

If we regard a polynomial p as defining a function R --> R, x |--> p(x), then p is differentiable, and

D: P_2 --> P_2, p |--> p' = dp/dx - defines a linear transformation.

Find the matrix of D with respect to the bases

(i) B in both the domain and co-domain

(ii) B in the domain and B' in the codomain.

(iii) B' in the domain and B in the codomain

(iv) B' in both the domain and codomain.

[/b]2. The attempt at a solution[/b]

I used 3x3 matrices and row reduction for B' to show that they are both bases, i.e. 3 pivot variables in reduced row form. B was just the identity matrix.

Its the second part that I don't understand. For me, the domains of B and B' are all real numbers, or have I already found the matrices of D in both domains by showing with row echelon matrices that B and B' are bases.

This is very confusing and my mind is now doing many laps around the same circuit.

Any clarification would be great.

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# Homework Help: Polynomial vector space

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