Exercise 3.10 Discussion

$\def\abs#1{\vert #1 \vert}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 3.10}$

Show that for any orthonormal basis $B = \{\ket{\beta_1}, \ket{\beta_2}, \dots, \ket{\beta_n}\}$ and vectors $\ket v = a_1\ket{\beta_1} + a_2\ket{\beta_2} + \dots + a_n \ket{\beta_n}$ and $\ket w = c_1\ket{\beta_1} + c_2\ket{\beta_2} + \dots + c_n \ket{\beta_n}$

a) the inner product of $\ket v$ and $\ket w$ is $\bar{c_1}a_1 + \bar{c_2}a_2 + \dots + \bar{c_2}a_2$, and
b) the length squared of $\ket v$ is $\abs{\ket v}^2 = \braket{v}{v} = \abs{a_1}^2 +\abs{a_2}^2 +\dots+ \abs{a_n}^2$.

Write all steps in Dirac's bra/ket notation.