Let F be a field with $\displaystyle q $ elements and let V be a vector space above F with dimension of $\displaystyle n $.
1. How many different basis do V have?
2. How many subspaces of dimension of $\displaystyle k $ do V have?
Let F be a field with $\displaystyle q $ elements and let V be a vector space above F with dimension of $\displaystyle n $.
1. How many different basis do V have?
2. How many subspaces of dimension of $\displaystyle k $ do V have?
It is not "who" but "how"...!נו באמת
Anyway: how many elements in V can be the first element in the basis? (all but the zero vector): how many elements can be the second element in a basis? (all but the scalar multiples of the first one, and there are q multiples like these), etc.
Try now to do the second part by yourself.
Tonio
No, not the whole answer but some highlights: there are $\displaystyle q^n -1 $ choices for the first vector (since zero cannot be part of a basis), then there are $\displaystyle q^n-q$ choices for the second vector (all the available vectors minus those that are a scalar multiple of the first one), then there are $\displaystyle q^n-q^2$ choices for the third one (all the possible vectors minus all the scalar multiples of the first two), etc.
All in, there are $\displaystyle (q^n-1)(q^n-q)(q^n-q^2)$...try to end the argument by yourself now.
Tonio
Ps ...הרבה יותר כיף לגלות את הפתרון בעצמך, אפילו אם זה קשה, מאשר לקבל את הכל מוכן