Exercise 2

Axiom (1)

\[ \begin{flalign*} \bra{C} \{ \ket{A} + \ket{B} \} &=\\ &= \begin{pmatrix} \gamma_1^* & \gamma_2^* & \dots \gamma_n^* \end{pmatrix} \left\{ \begin{pmatrix} \alpha_1\\ \alpha_2\\ \vdots \\ \alpha_n \end{pmatrix} + \begin{pmatrix} \beta_1\\ \beta_2\\ \vdots \\ \beta_n \end{pmatrix} \right\}\\ &= \begin{pmatrix} \gamma_1^* & \gamma_2^* & \dots \gamma_n^* \end{pmatrix} \begin{pmatrix} \alpha_1 +\beta_1 \\ \alpha_2 + \beta_2\\ \vdots \\ \alpha_n+\beta_n \end{pmatrix}\\ &= \gamma_1^* (\alpha_1 + \beta_1) + \gamma_2^*(\alpha_2+\beta_2) + \dots + \gamma_n^* (\alpha_n + \beta_n)\\ &= \gamma_1^* \alpha_1 + \gamma_2^* \alpha_2 + \dots + \gamma_n^* \alpha_n + \gamma_1^* \beta_1 + \gamma_2^* \beta_2 + \dots + \gamma_n^* \beta_n \\ &= \braket{C|A} + \braket{C|B} \end{flalign*} \]

Axiom (2)

\[ \begin{flalign*} \braket{B|A} &=\\ &= \begin{pmatrix} \beta_1^* & \beta_2^* \dots \beta_n^* \end{pmatrix} \begin{pmatrix} \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_n \end{pmatrix} \\ &= \beta_1^* \alpha_1 + \beta_2^* \alpha_2 + \dots + \beta_n^* \alpha_n \\ &= \alpha_1 \beta_1^* + \alpha_2 \beta_2^*+ \dots + \alpha_n \beta_n^*\\ &= (\alpha_1^* \beta_1 + \alpha_2^* \beta_2+ \dots + \alpha_n^* \beta_n)^*\\ &= \left\{ \begin{pmatrix} \alpha_1^* & \alpha_2^* \dots \alpha_n^* \end{pmatrix} \begin{pmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_n \end{pmatrix} \right\}^*\\ &= \braket{A|B}^* \end{flalign*} \]