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?

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- Dec 2nd 2009, 02:44 PMAlso sprach ZarathustraField and vector space
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? - Dec 2nd 2009, 03:13 PMtonio

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 - Dec 2nd 2009, 03:28 PMAlso sprach Zarathustra
Shalom! I am mitbaesh beatzmi...

Can you please write the whole answer for the question.

1. q^n-1 ?

2.? - Dec 2nd 2009, 03:32 PMAlso sprach Zarathustra
p.s:

What is MHF expert mean? - Dec 2nd 2009, 07:15 PMtonio
- Dec 2nd 2009, 07:21 PMtonio

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 - Dec 3rd 2009, 12:47 AMAlso sprach Zarathustra
And what about 2?

- Dec 3rd 2009, 03:04 AMDefunkt
- Dec 3rd 2009, 03:59 PMtonio
- Dec 3rd 2009, 04:09 PMDefunkt
יפה יפה... אני מניח שכבר סיימת תואר ראשון?