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What you really want to know are the definitions of the $\sigma_i$ --- these are the Pauli matrices: $$\sigma_x = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \qquad \sigma_y = \begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix} \qquad \sigma_z = \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}$$ Hopefully you can see now how the equation ...
As stated, $\mathbf{n}$ is a unit vector and $n_x$, $n_y$ and $n_z$ are its cartesian components. $\mathbf{n}$ is just a vector pointing in an arbitrarily direction with magnitude 1. Taking $\mathbf{n} \cdot \mathbf{\sigma}$, we have \mathbf{n} \cdot \mathbf{\sigma} = n_x\sigma_x + n_y \sigma_y + n_z \sigma_z \\ = n_x \left(\begin{array}{cc} ...