Exercise 1

Exercise 1 a)

At each step, the axiom used is given:

\[\begin{flalign*} \{ \bra{A} + \bra{B} \} \, \ket{C} &=\\ &\stackrel{\textrm (2)}{=} \left[ \bra{C} \, \{ \ket{A} + \ket{B} \} \right]^* \\ &\stackrel{\textrm (1)}{=} \braket{C|A}^* + \braket{C|B}^* \\ &\stackrel{\textrm (2)}{=} \braket{A|C} + \braket{B|C} \end{flalign*}\]

Exercise 1 b)

Look at axiom (2) for the case \(\bra{B} = \bra{A}\). We get \[ \braket{A|A} = \braket{A|A}^* \Rightarrow \braket{A|A} \in \mathbb{R} \]

If a complex number is equal to its own complex conjugate, its imaginary part is equal to zero: \[ z = z^* \] \[ x + \mathrm{i}y = x - \mathrm{i}y \Rightarrow \mathrm{i}y = - \mathrm{i}y \Rightarrow y = 0 \]

Hence, the number is real. You can visualize this easily in the complex number plane: The complex conjugate of a number is achieved by mirroring at the \(x\)-axis. All numbers that match their mirror image are located on the \(x\)-axis, these are the real numbers.