Linear operator Definition and 110 Threads

Homework Statement If I e.g. want to find the kernel and range of the linear opertor on P_3: L(p(x)) = x*p'(x), then we can write this as L(p'(x)) = x*(2ax+b). What, and why, is the kernel and range of this operator? The Attempt at a Solution The kernel must be the x's where L(p'(x))...

Q: Suppose V is a finite dimensional inner product space and T:V->V a linear operator. a) Prove im(T*)=(ker T)^(|) b) Prove rank(T)=rank(T*) Note: ^(|) is orthogonal complement For this question, I don't even know how to start, so it would be nice if someone can give me some hints. Thank...

Q) Let V be an inner product space and T:V->V a linear operator. Prove that if T is normal, then T and T* have the same image. (i.e. imT=imT*) My Attempt: <T(v),T(v)> =<T*T(v),v> =<TT*(v),v> =<T*(v),T*(v)> =>||T(v)|| = || T*(v)|| But this doesn't seem to help... Thanks!

Definition from my textbook: For each linear operator T on a inner product space V, the adjoint of T is the mapping T* of V into V that is defined by the equation <T*(v),w> = <v,T(w)> for all v, w E V. My instructor defined it by <T(v),w> = <v,T*(w)> and he said that these 2 definitions are...

I understand the definition of trace and linear operator individually but I don't seem to understand as to what does it mean by trace of a linear operator on a finite dimensional linear space. What I have found out is that trace of a linear operator on a finite dimensional linear space is the...

Consider the linear operator T on \mathcal{C}^2 with the matrix \bmatrix 2 && -3\\3 && 2 \endbmatrix in the standard basis. With the basis vectors \frac{1}{\sqrt{2}} \bmatrix i \\ 1 \endbmatrix, \quad \frac{1}{\sqrt{2}} \bmatrix -i \\ 1 \endbmatrix this operator can be written \bmatrix 2+3i...

Let be L and G 2 linear operators so they have the same set of Eigenvalues, then: L[y]=-\lambda _{n} y and G[y]=-\lambda _{n} y then i believe that either L=G or L and G are related by some linear transform or whatever, in the same case it happens with Matrices having the same...

let be the linear operator: (Hermitian ??) L = -i(x\frac{d}{dx}+1/2) then the "eigenfunctions" are y_{n} (x)=Ax^{i\lambda _{n} -1/2 then my question is how would we get the energies imposing boundary conditions? (for example y(0)=Y(L)=0 wher L is a positive integer )...:smile: :smile:

How do I go about proving the following partial derivitive is a linear operator? d/dx[k(x)du/dx)]

Let T: R^3 -> R^3 be a linear operator and T(x;y;z)= (x -y + 3z;2x-y+8z;3x -5y+5z). Find the rule, for the linear operator S: R^3 -> R^3 such that ker S= I am T and I am S=Ker T. I'm not really sure how I should start this problem. Also I would like to know if I can assume S is the...