Understanding Quantum Physics Identities

[kennalj65]

Hey, so today for our quantum physics class we were supposed to go through these identities, $|+_x > = \frac{1}{2^{0.5}} (|+> + |->)$$|-_x > = \frac{1}{2^{0.5}} (|+> - |->)$
$|+_y > = \frac{1}{2^{0.5}} (|+> + i|->)$
$|-_y > = \frac{1}{2^{0.5}} (|+> - i|->)$where |+ (x)> would represent spin up in the x direction for example, and |+> simply denotes spin up in (I believe) the z direction.
now I couldn't make sense of any of it, had no idea where they came from and what the proof is, I came home and googled it and noticed a strong resemblance to the pauli matrices (which I have discovered as of half an hour ago), so I'm hoping someone could enlighten me as to what this means and where it comes from.
Thanks in advance

 

[vanhees71]

The Pauli matrices are the matrix elements of the operators
$$\hat{\sigma}^j=\frac{2}{\hbar} \hat{s}^j,$$
where $\hat{s}_j$ are the spin components of a spin-1/2 particle.

The Pauli matrix are the matrix elements with respect to the eigenbasis of $\hat{\sigma}^3$,
$$\hat{\sigma}^3 |k \rangle=k |k \rangle, \quad k \in \{-1,1 \},$$
where the choice of the 3-component is conventional. The Pauli matrices are given by
$${\sigma^{j}}_{kl}=\langle{k}|\hat{\sigma}^j|l\rangle, \quad k,l \in \{-1,1 \}.$$

You can easily calculate the Pauli matrices by making use of the raising- and lowering operators
$$\hat{\sigma}^{\pm} = \hat{\sigma}^{1} \pm \mathrm{i} \hat{\sigma}^{2}.$$
Just look in your textbook, how those operate on the basis vectors!