Direct Product of Two Spin-Up Vectors: What Is It?

[dyn]

Hi
I have just started looking at direct products and came across the following which i don't understand :

the direct product of two spin -up vectors = | 1 > which is in a bigger vector space

I don't understand how the direct product is | 1 > ? and in this case is it always a bigger vector space or the same size as the original vector spaces ?

 

[Khashishi]

That's just sloppy notation. If you are trying to understand direct product, it's basically just making an ordered pair of the two vectors.
|up1> x |up2> = |up1, up2>
Now clearly, the total z angular momentum of |up1, up2> is 1hbar so we can also label this state |1>. But that's just notation, not math.