Wavefunction in Dirac notation

Homework Statement

For the infinite square well, a particle is in a state given by $\psi = \frac{1}{\sqrt 2}(\psi_1 + \psi_3)$ , where $\psi_1$ and $\psi_3$ are energy eigenstates (ground state and the second excited state, respectively).

Represent this state as a column matrix $\psi>$ in the energy basis and x basis. You may use your knowledge of the solutions of the infinite square well from before, obtained in the x basis. State with the help of mathematical equations how you would find the column matrix in k basis.

Homework Equations

I know that in the x-space, the column matrix representation of basis vector is |x> and the components of a state vector $\psi$ is $<x|\psi>$. And likewise, replace 'x' with 'k' for the k-space basis.

The Attempt at a Solution

Is writing $\psi = \frac{1}{\sqrt 2}(\psi_1 + \psi_3)= \frac{1}{\sqrt 2}(<i|\psi_1>+<i|\psi_3>)$ permitted? If not, can you please point me in the right directions? I have Gritffiths Introduction to QM book so any reference to that I can get hold of. Thanks.