# 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.

rem ipsum dolor sit amet, consectetur adipiscing elit. Nam

a. Fusce dui lectus, congue vel laoreet ac, dictum vitae odio. Donec aliquet. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Nam lacinia pulvinar tortor nec facilisis. Pellentesque dapibus efficitur laoreet. Nam risus ante, dapibus a molestie consequat, ul
**487,593 students got unstuck** by Course

Hero in the last week

**Our Expert Tutors** provide step by step solutions to help you excel in your courses