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**Theorem 5.3:**Let A be in [tex]M_n_x_n(F)[/tex].

(a) The characteristic polynomial of A is a polynomial of degree n with leading coefficient [tex](-1)^n[/tex].

(b) A has at most n distinct eigenvalues.

*Note: The theorem can be proved by a straightfoward induction arguement.*

**Question:**Can someone help with the proofs? For part (b), I understand there can be at most n distinct eigenvalues, since the dimension of the matrix is the same as the number of elements along the diagonal. For this reason, there can be at most n distinct eigenvalues. But for (b), does the proof require induction also, or is the text simply encouraging induction for part (a)?

**One last easy question:**Let T be the linear operator on [tex]P_2(R)[/tex] defined by [tex]T(f(x)) = f(x) + (x + 1)f'(x)[/tex], let B be the standard ordered basis for [tex]P_2(R)[/tex], and let A = [tex][T]_B[/tex]. Then,

A = { (1, 0, 0), (1, 2, 0), (0, 2, 3) } This is a matrix with each paranthesis being column vectors.

In this example, i know B = { [tex]1, x, x^2[/tex] } is an ordered basis for [tex]P_2(R)[/tex]. So we plug the first element into the equation [tex]T(f(x)) = f(x) + (x + 1)f'(x)[/tex], and then plug x, then finally x^2. But for some reason I don't know how to evaluate each equation to get the respective column vectors above. In particular what is f(1), or what is [tex]f(x^2)[/tex]?

Thanks so much,

JL