- #1

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- Homework Statement
- Let ##V## be the Linear space of all real polynomials of degree ##\leq3##. Let D denote the differentiation operator and let ##T : V \to V## be the Linear transformation which maps ##p(x)## onto ##x p'(x)##.

Let ##W## be the image of ##V## under ## TD##. Find the bases for V and W relative to which the matrix of ##TD## is in diagonal form.

- Relevant Equations
- See the main body, please

First of all, it is clear that we can find such a bases (the theorem is given in almost all of the books, but if you want to share some insight I shall be highly grateful.)

We can show that ##W## will be the set of all real polynomials with degree ##\leq 2##. So, let's have ##\{1,x,x^2\}## as the basis for ##W##.

We want this matrix (having this idea that V is four dimensional):

$$

\begin{bmatrix}

1 &0&0&0\\

0&1&0&0\\

0&0&1&0

\end{bmatrix}

$$

Let the basis for ##V## be ##\{e_1,e_2,e_3,e_4\}##. Then, ##e_1 = c_1 + c_2x +c_3x^2 + c_4x^3##, we have

$$

TD (e_1)= 1 $$

Which implies

$$

6c_4 x^2 + 2c_3x -1=0$$

But I cannot solve c's. Now, I seek for your gentle guidance.

We can show that ##W## will be the set of all real polynomials with degree ##\leq 2##. So, let's have ##\{1,x,x^2\}## as the basis for ##W##.

We want this matrix (having this idea that V is four dimensional):

$$

\begin{bmatrix}

1 &0&0&0\\

0&1&0&0\\

0&0&1&0

\end{bmatrix}

$$

Let the basis for ##V## be ##\{e_1,e_2,e_3,e_4\}##. Then, ##e_1 = c_1 + c_2x +c_3x^2 + c_4x^3##, we have

$$

TD (e_1)= 1 $$

Which implies

$$

6c_4 x^2 + 2c_3x -1=0$$

But I cannot solve c's. Now, I seek for your gentle guidance.

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