Why not just complex conjugate bras and kets instead of Hermitian conjugate?

Question

I read that one equation involving bras, kets and operators, implies another equation (its transpose conjugate), analogous to how one equation involving complex numbers implies its complex conjugate equation.

I'm not seeing the need of the transpose though. Why not just only do the conjugate? As an example:

Let's consider an inner product equation $\langle A|B\rangle=z$, where $z$ is a complex number, $\langle A|$ is the bra $(a,b,c)$, $|B\rangle$ is the ket $(d,e,f)$.

This equation basically expresses this relation between a bunch of complex numbers:

$$ad+be+cf=z$$

If we simply conjugate every complex number inside the bras and kets, we get this implied relation:

$$a^*d^*+b^*e^*+c^*f^*=z^*$$

The transpose conjugate equation, $\langle B|A\rangle=z^*$, also expresses the same relation as above but in a notationally flipped manner.

So what's the need of the transpose?

Answer 1

Seeing as the following might help:

$$|\psi\rangle\rightarrow \begin{pmatrix} a\\ b \end{pmatrix} $$ $$\langle \psi|\rightarrow \begin{pmatrix} a^*&b^* \end{pmatrix} $$ In this way the inner produce can be written as $$\langle \phi|\psi\rangle \rightarrow \begin{pmatrix} c^*&d^* \end{pmatrix} \begin{pmatrix} a\\ b \end{pmatrix}=ac^*+bd^* $$ $$\langle \psi|\phi\rangle \rightarrow \begin{pmatrix} a^*&b^* \end{pmatrix} \begin{pmatrix} c\\ d \end{pmatrix}=ca^*+db^* =(ac^*+bd^*)^*$$ As expected. That explains the need of taking transpose. The reason for taking conjugate is that we want $\langle \psi|\psi\rangle $ to be positive which represent the length of a vector.