In a three-dimensional complex linear vector space, consider two vectors |ψi = -|a1i + i|a3i and |φi = 2|a1i + 3i|a2i + 4|a3i, where |a1i, |a2i, and |a3i are orthonormal basis vectors spanning the vector space.
(a) Calculate the inner product hψ|φi = C and outer product |ψihφ| = P. Comment on the nature of C and P, whether they are ket-vectors, bra-vectors, complex numbers, or operators.
(b) Compute the following expression P(3|ψi + 2i|φi).
(c) Express the |ψi, |φi, and P in terms of their corresponding matrix form and repeat parts (a) and (b) using matrices.