Dirac's Notation

In quantum mechanics, we can describe "pure" quantum states using an n-dimensional vector or rank 1 tensor, this is called ket and is represented with: $$  \vert \Psi  \rangle  =  \begin{pmatrix} \alpha_1 \\ \vdots \\ \alpha_n \end{pmatrix} $$

We can form the corresponding bra (dual space) transposing the ket and complex conjugating every component: $$  \langle \Psi  \vert  =  \begin{pmatrix} \alpha_1^* & \cdots & \alpha_n^*  \end{pmatrix} $$ At first instance we can define some products using bras and kets, the inner product, outer product and tensorial product.

Inner product:  \begin{equation*} \langle \Psi  \vert \Psi  \rangle  =  \begin{pmatrix} \alpha_1^* & \cdots & \alpha_n^* \end{pmatrix} \begin{pmatrix} \alpha_1 \\ \vdots \\ \alpha_n \end{pmatrix} =  \alpha_1^*  \alpha_1 + \cdots \alpha_n^*  \alpha_n =  \vert \alpha_1 \vert ^2 + \cdots + \vert \alpha_n \vert ^2  \end{equation*}   

Outer product: \begin{align*} \vert  \Psi \rangle \langle  \Psi  \vert  &=   \begin{pmatrix} \alpha_1 \\ \vdots \\ \alpha_n \end{pmatrix} \begin{pmatrix} \alpha_1^* & \cdots & \alpha_n^* \end{pmatrix} =  \begin{pmatrix}  \alpha_1  \alpha_1^* &  \alpha_1  \alpha_2^*  & \cdots &  \alpha_1  \alpha_n^* \\  \alpha_2  \alpha_1^* &  \alpha_2  \alpha_2^*  & \cdots &  \alpha_2  \alpha_n^* \\ \vdots & \vdots & \ddots & \vdots \\  \alpha_n  \alpha_1^* &  \alpha_n  \alpha_2^*  & \cdots &  \alpha_n  \alpha_n^* \end{pmatrix} \\ \\  &=   \begin{pmatrix}  \vert \alpha_1 \vert ^2 &  \alpha_1  \alpha_2^*  & \cdots &  \alpha_1  \alpha_n^* \\  \alpha_2  \alpha_1^* &  \vert \alpha_2 \vert ^2  & \cdots &  \alpha_2  \alpha_n^* \\ \vdots & \vdots & \ddots & \vdots \\  \alpha_n  \alpha_1^* &  \alpha_n  \alpha_2^*  & \cdots & \vert \alpha_n \vert ^2  \end{pmatrix} \end{align*}

Tensorial product: $ \vert \alpha \rangle \otimes \vert \beta \rangle \equiv \vert \alpha \rangle \vert \beta \rangle \equiv  \vert \alpha , \beta \rangle \equiv  \vert \alpha \beta \rangle $. Supposing 2-dimensional kets $ \vert \alpha \rangle = \big(\begin{smallmatrix} \alpha_1 \\ \alpha_2 \end{smallmatrix}\big) $ and $\vert \beta \rangle = \big(\begin{smallmatrix} \beta_1 \\ \beta_2 \end{smallmatrix}\big) $ we can define the tensorial product as:

\begin{align*} \vert \alpha \rangle \otimes \vert \beta \rangle  &=   \begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix} \otimes \begin{pmatrix} \beta_1 \\ \beta_2 \end{pmatrix} =  \begin{pmatrix} \alpha_1 \begin{pmatrix} \beta_1 \\ \beta_2  \end{pmatrix} \\ \alpha_2  \begin{pmatrix} \beta_1 \\ \beta_2  \end{pmatrix} \end{pmatrix} = \begin{pmatrix} \alpha_1 \beta_1 \\  \alpha_1 \beta_2  \\  \alpha_2 \beta_1 \\  \alpha_2 \beta_2  \end{pmatrix}  \end{align*}