Can the wave-function of any particle in any basis be written as a matrix?

Can the wave-function of any particle in any basis be written as a matrix?

If no, how can we explain this, where the Hamiltonian $$H$$ in U$Ψ=e^{iHt}$ is a QM operator that can be written as a linear transformation therefore a matrix. And if we take the exponential matrix of H, which gives us another matrix. So surely, we can write Ψ as a matrix. Right??

Thus in a given basis $$\vec{v_1}$$ and $$\vec{v_2}$$ the state $$\vec{v}=a\vec{v_1}+b\vec{v_2}$$ can be represented as a vector $$(a\;\; b)$$.