1. ## Normal, self-adjoint, or neither linear operator

In $\displaystyle P_{2}( \mathbb {R} )$, let T be defined by T(f) = f', where $\displaystyle <f,g> = \int _{0}^{1} f(t)g(t)dt$

Is T normal, self-adjoint, or neither?

In $\displaystyle M_{2x2} ( \mathbb {R} )$, define T by $\displaystyle T(A) = A^t$

Same problem here.

In $\displaystyle P_{2}( \mathbb {R} )$, let T be defined by T(f) = f', where $\displaystyle <f,g> = \int _{0}^{1} f(t)g(t)dt$

Is T normal, self-adjoint, or neither?

In $\displaystyle M_{2x2} ( \mathbb {R} )$, define T by $\displaystyle T(A) = A^t$

Same problem here.

I don't know how to find T* from here.
The adjoint operator is given by $\displaystyle \langle T^*f,g\rangle = \langle f,Tg\rangle$. If Tg = g', and the inner product is given by integration over the unit interval, then $\displaystyle \langle f,Tg\rangle = \int _{0}^{1} f(t)g'(t)\,dt$. You want to write this in the form $\displaystyle \int _{0}^{1} ???g(t)\,dt$, where "???" will be T*(f).

3. Can I say $\displaystyle T^*(f) = \frac {f(t)g(t)}{g'(t)}$?

And for the second one, I have:

$\displaystyle <T(A),B> = <A,T^*(B)>$

$\displaystyle <T(A),B>=<A^t,B>=tr(B^t,A^t)$

and $\displaystyle <A,T^*(B)>=<A,???>=tr(???^t,A)$

Then I'm bit lost, how I can find something that would fit ???

4. For the second one, you're almost there. If the inner product is given by $\displaystyle \langle A,B\rangle = \text{tr}(B^{\textsc T}A)$, and $\displaystyle T(A) = A^{\textsc T}$, then $\displaystyle \langle T(A),B\rangle = \text{tr}(B^{\textsc T}A^{\textsc T})$. But this is the same as $\displaystyle \langle A,T(B)\rangle$ (since $\displaystyle \text{tr}(M) = \text{tr}(M^{\textsc T})$). So T is selfadjoint.

The first one is a good deal more complicated than I first thought, and I wonder if there is something wrong with the question. My original idea was to use integration by parts to express $\displaystyle \textstyle\int fg'$ in terms of $\displaystyle \textstyle\int f'g$. But when you do the integration by parts, with the limits 0 and 1, you actually get $\displaystyle \langle f',g\rangle = f(1)g(1)-f(0)g(0) - \langle f,g'\rangle$. The awkward terms f(1)g(1)-f(0)g(0) prevent you from getting a neat expression for the adjoint of the differentiation operator. That seems to have the effect of making this question unreasonably difficult.

5. I asked my professor, and he said we should use this theorem:

T is normal if and only if there exist an orthonormal basis for V consisting of eigenvectors of T.

So, to find eigenvectors of T, I have $\displaystyle T(f)=f'= \lambda f$ So I need to find some polynomials f, are there any?

the standard ordered basis of V is $\displaystyle \{ 1,x,x^2 \}$

So the matrix $\displaystyle [T]_{ \beta } = \begin{pmatrix} 0 && 0 && 0 \\ 0 && 1 && 2 \\ 0 && 0 && 0 \end{pmatrix}$

So the eigenvalue of T is 0,1.

Well, then any polynomial to the first power would satisfy T, right?

In $\displaystyle P_{2}( \mathbb {R} )$, let T be defined by T(f) = f', where $\displaystyle <f,g> = \int _{0}^{1} f(t)g(t)dt$

Is T normal, self-adjoint, or neither?
Just to be clear about this, I assume that $\displaystyle P_{2}( \mathbb {R} )$ means polynomials of degree at most 2 over the real numbers.

I asked my professor, and he said we should use this theorem:

T is normal if and only if there exist an orthonormal basis for V consisting of eigenvectors of T.
Interesting suggestion. I hadn't thought of doing it that way. There's a snag, though. That theorem works if the scalars are the complex numbers, but you have to be careful how to use it if the scalars are the reals. For example, the matrix $\displaystyle \begin{bmatrix}0&1\\-1&0\end{bmatrix}$ is normal, but has no real eigenvalues or eigenvectors.

To use the theorem here, you must first show that the complex space $\displaystyle P_{2}( \mathbb {C} )$ does not have a base consisting of eigenvectors of T, and then deduce that since T is not normal as an operator on that space it must also fail to be normal when we restrict to real scalars.

So, to find eigenvectors of T, I have $\displaystyle T(f)=f'= \lambda f$ So I need to find some polynomials f, are there any?
the standard ordered basis of V is $\displaystyle \{ 1,x,x^2 \}$
So the matrix $\displaystyle [T]_{ \beta } = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{pmatrix}$ Not quite; the 1 should be at the bottom of the middle column.