State vector construction method vs observable in "Quantum Mechanics The Theoretical Minimum" In L. Susskind's book "Quantum Mechanics The Theoretical Minimum, spin is used as a vehicle to explain the effect of an observable in three orthogonal diagrams, leading to the creation of the Pauli Spin matrices. My question is the following specific aspect of the reasoning leading to the Pauli Spin matrices. 
The notation used is: Along the $z$-axis, we measure the spin states $\lvert u\rangle$ 'up' and $\lvert d\rangle$ 'down'. Along the $x$-axis, we measure $\lvert l\rangle$ 'left' and $\lvert r\rangle$ 'right', and along the $y$-axis, we measure $\lvert i\rangle$ 'in' and $\lvert o\rangle$ 'out'.
We have
$$|r\rangle = \frac{1}{\sqrt{2}}|u\rangle+\frac{1}{\sqrt{2}}|d\rangle\tag{1}$$
$$ \sigma_z = 
\begin{pmatrix}
1 & 0 \\
0 & -1\\
\end{pmatrix}
$$
$$\sigma_{z}|r\rangle = \frac{1}{\sqrt{2}}|u\rangle-\frac{1}{\sqrt{2}}|d\rangle\tag{2}$$
Equation (1) is derived, see book page 41, equation 2.5, Section 2.3 by using the following setup:


*

*Apparatus A prepares the spin state to be $|r\rangle$

*Apparatus A is then rotated to measure $\sigma_{z}$
However this "looks" to be doing the same as the left hand side of equation (2), but the right hand side of equation (2) [derived on page 82] has a minus sign that isn't there in equation (1). So why is there a difference?
 A: After the apparatus measured the spin along the +x axis, the particle has the state $|r\gt$. 
Applying the operator $ \sigma_z$ doesn't modify that state. It is not a measurement (yet). So:
$$\sigma_{z}|r\gt = \frac{1}{\sqrt{2}}|u\gt-\frac{1}{\sqrt{2}}|d\gt$$
is just the acting of 
$$ \sigma_z = 
\begin{pmatrix}
1 & 0 \\
0 & -1\\
\end{pmatrix}
$$
over the state $|r\gt$. It is a multiplication of that vector by that matrix. The minus sign come from the matrix.
The meaning of that operation is to be able to calculate the probability and expected value of an eventual measurement along +z orientation.  
A: In page 41, we are only told that measuring $|r\rangle$ gives us $|u\rangle$ and $|d\rangle$ with equal probabilities. Here we are using the measurement of $\sigma_z$ of an electron in $|r\rangle$ state to determine it in terms of $|u\rangle$ and $|d\rangle$. We are only looking at probabilities here. Not probability amplitudes. Thus when forming the state we have the freedom to call either $$|r\rangle=\frac{1}{\sqrt{2}}|u\rangle \pm \frac{1}{\sqrt{2}} |d\rangle$$ as the state. 
We chose to call the plus state as $|r\rangle$. 
However in page 82, we are considering the action of $\sigma_z$ on the state $|r\rangle$ at the level of amplitudes. And since we have already decided on what the state $|r\rangle$ is, we can do this calculation which turns out to be $\frac{1}{\sqrt{2}}|u\rangle - \frac{1}{\sqrt{2}} |d\rangle$. 
