Transpose

We just learned what ket $\left\lvert x \right\rangle$ means — a $N\times 1$ vector. Let that ALSO be called a ket. If we flip it horizontally, we get a bra $\left\langle x \right\rvert$ .

If a ket is

\begin{gather*} \left\lvert x \right\rangle=\begin{pmatrix} 1+3i \\ 2+5i \\ 7 \end{pmatrix}\quad \left\lvert \psi \right\rangle=\begin{pmatrix} 9 \\ 9 \\ 0 \\ 0 \\ 0 \end{pmatrix}\quad\left\lvert \phi \right\rangle=\begin{pmatrix} 0 \\ 1 \end{pmatrix} \end{gather*}

Then it’s counterpart is a bra

\begin{gather*} \left\langle x \right\rvert=\begin{pmatrix} 1-3i & 2-5i & 7 \end{pmatrix}\quad \left\langle \psi \right\rvert=\begin{pmatrix} 9 & 9 & 0 & 0 & 0 \end{pmatrix}\quad\left\langle \phi \right\rvert=\begin{pmatrix} 0 & 1 \end{pmatrix} \end{gather*}

You swap the columns for the rows (transpose!) and make every Element the Conjugate of itself. It becomes a $1\times N$ matrix instead.

Let this weird transformation be known as a conjugate transpose, denoted as

\begin{gather*} \left\langle v \right\rvert=(\left\lvert v \right\rangle)^+ \end{gather*}