A few problems with Quantum Mechanics

I am having trouble understanding a few problems regarding Quantum Mechanics. I believe that those are mathematical problems really, but I have tried to read through all kinds of Linear Algebra books and can't seem to figure it out.

My problems are:
Vectors I1> and I2> create the orthonormal basis.
Operator O is:
O=a(l1><1l-l2><2l+il1><2l-il2><1l), where a is a real number.
Find the matrix representation of the operator in the basis I1>,I2>. Find eigenvalues and eigenvectors of the operator. Check if the eigenvectors are orthonormal.

Here, I am having trouble with the first part. I am not sure how to construct that matrix and am really not familiar with this notation of the operators. Once I got the matrix, I believe I would know how to calculate eigenvectors and eigenvalues.

The second problem is quite similar, but I also don't know how to solve it.
Eigenvectors of the Hamiltonian, with appropriate energies are:
lv1> = (l1>+l2>+l3>)/√3
lv2> = (l1>-l3>)/√2
lv3> = (2l2>-l1>-l3>)/√6

E1=α+2β
E2=α-β
E3=α-β

Find the matrix representation of the Hamiltonian in the basis l1>,l2>,l3> . If in t=0, system is in the state l1>, how does the wave function look in the time t?