Number of distinct base

[infinityQ]

Hello, everyone.

I would like to get some concrete examples of the number of distinct bases of k-dimensional vector space W over a finite field F with q elements.

The formula for the number of distinct bases of W is ( p 412 Dummit )
(q^k - 1 )(q^k - q)(q^k - q^2)...(q^k - q ^(k-1) )

I am having a hard time finding an example of F with 2 elements and W be 2-dimensional vector space. According to the formula, 6 distinct bases of W should be acquired.

I will appreciate if anyone shows these 6 distinct bases of W.

Thanks in advance.

 

[OfirD]

take the $$Z_2$$ field and the vector space $$(Z_2)^2$$ meaning the vectors are { (0,0), (0,1), (1,0), (1,1) }
any two of the three last vectors are a base for the vector space

 

[infinityQ]

6 distinct bases of W might be

(0,1)(1,0)
(0,1)(1,1)
(1,0)(0,1)
(1,0)(1,1)
(1,1)(0,1)
(1,1)(1,0)

But my question is second one and fifth one, for instance, does not look distinct. So it seems to me just four distinct bases exists for W.

Any suggestion?

 

[morphism]

That formula counts ordered bases and not just any bases. So your second and fifth bases are actually different in this sense.