Linear Transformations using polynomials

[mbud]

Homework Statement

Let P3 be the space of all polynomials (with real coefficients) of degree at most 3. Let
D : P3 -> P3 be the linear transformation given by taking the derivative of a polynomial.
That is
D(a + bx + cx2 + dx3) = b + 2cx + 3dx2:
Let B be the standard basis {1; x; x2; x3} of P3.
(a) Find the matrix MD of D with respect to the standard basis.
(b) Explain, without doing any matrix calculations, why (MD)4 = 0.

Homework Equations

The Attempt at a Solution

i know it may be a simple question but i don't even know where to begin
but in a attempt is
MD = [1 1 1 1;
0 2 3 0]

 

[tiny-tim]

Welcome to PF!

Hi mbud! Welcome to PF!

(try using the X² tag just above the Reply box )

Let P3 be the space of all polynomials (with real coefficients) of degree at most 3. Let
D : P3 -> P3 be the linear transformation given by taking the derivative of a polynomial.
That is
D(a + bx + cx2 + dx3) = b + 2cx + 3dx2:
Let B be the standard basis {1; x; x2; x3} of P3.
(a) Find the matrix MD of D with respect to the standard basis.
(b) Explain, without doing any matrix calculations, why (MD)4 = 0.

To find M, you only need the effect of M on the four basis elements. Sooo …

Hint: what is x² as a vector? what is Dx² as a vector?

 

[HallsofIvy]

More specifically, since the standard basis is 1, x, x², and x³, take the derivative of each and write it in terms of those. The coefficients give each column of the matrix.

 

[mbud]

Thanks, for the help, but i still don't know how to explain (MD)^4 = 0
I can show it using matrices, but when it comes to words, I am flabbergasted.

 

[tiny-tim]

Thanks, for the help, but i still don't know how to explain (MD)^4 = 0
I can show it using matrices, but when it comes to words, I am flabbergasted.

Hint: if V is the vector form of a polynomial ax³ + bx² + cx + d, what is (MD)⁴V the vector of?

 

[matt grime]

Differentiation does what to the degree of a polynomial? This is nothing to do with vectors, or vector spaces, or matrices - sometimes you just have to look at things and think for a little bit.