Let V ={(a1,, an)|aiF2} be an n -dimensional vector space over the field of 2 elements F2. A basis of V is a sequence of n linearly independent
Question

# Let V ={(a1,···, an)|ai∈F2} be an n -dimensional vector space over

the field of 2 elements F2. A basis of V is a sequence of n linearly independent vectors in V. In this exercise, we will count the number of bases of V.

(1) To count the number of bases, let's recall how to construct a basis. The first step is to pick a non-zero element v1. How many choices of v1 do we have?

(2) Suppose we have found the first k-1 vectors in the basis. The k-th vector in the basis can be any vector that is linearly independent of the first k−1
vectors. This is equivalent to say vk/∈Span{v1, v2...., vk−1}.1 So how many choices of vk do we have?

(3) Multiply the numbers of choices for each v 1, v2,···, vn together, we get the number of bases of V. Write down its formula.

(4) Verify this formula when n=2 , by finding all bases of V in that case.

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