Expressing Spin State |r> As Linear Superposition of |u> and |d>: Basic Linear Algebra?

Background This question, from Quantum Mechanics: The Theoretical Minimum started with the following assumption(?)

$$|r\rangle = \frac{1}{\sqrt 2}|u\rangle + \frac{1}{\sqrt 2}|d\rangle$$

I'm now attempting to derive(?) the above based on the initial information given, which I think, based on the paragraph immediately above, is as follows:

$$|r\rangle = x|u\rangle + y|d\rangle$$ and $$\langle r|u\rangle \langle u|r\rangle = \frac{1}{2}$$ and $$\langle r|d\rangle \langle d|r\rangle = \frac{1}{2}$$

Questions

1. Is my assumption about the initial information correct?
2. If #1 is 'yes' would solving the above be covered in basic linear algebra?

NOTE: Please do not provide a solution to the set of equations.

• Yes and yes. This is exactly what you learn in linear algebra when you learn about basis decomposition. Apr 11 '16 at 21:20
• @knzhou Thanks. And I was afraid you were going to say that. Apr 12 '16 at 13:21