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Bilzero column vectors (two entries long) with entries in Z/3. Then, given a nonzero vector, you need to figure out how many vectors are not multiples of it. (c) If you succeed at (b), you probably can give me a formula for the number of elements of GL(2, Z/p) for any prime p. Try it!

Bilzero column vectors (two entries long) with entries in Z/3. Then, given a nonzero vector, you need to figure out how many vectors are not multiples of it. (c) If you succeed at (b), you probably can give me a formula for the number of elements of GL(2, Z/p) for any prime p. Try it!

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 3AEXP
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I need help with part c
2 ok (b) There are a lot of 2 x 2 matrices with entries in Z/3 and probably you don't want to list all of them. It turns out that there is a different way to figure out how many elements are in GL(2, Z/3). You might remember that a 2 x 2 matrix with entries in R is invertible if the first column is not the zero vector and if the second column is not in the span of the first column, which is to say, not a multiple of the first column. The same works for matrices with entries in Z/p (take this on faith). Your job is to find the number of elements in GL(2,Z/3). You first need to figure out the number of nonzero column vectors (two entries long) with entries in Z/3. Then, given a nonzero vector, you need to figure out how many vectors are not multiples of it. (c) If you succeed at (b), you probably can give me a formula for the number of elements of GL(2, Z/p) for any prime p. Try it!
Transcribed Image Text:2 ok (b) There are a lot of 2 x 2 matrices with entries in Z/3 and probably you don't want to list all of them. It turns out that there is a different way to figure out how many elements are in GL(2, Z/3). You might remember that a 2 x 2 matrix with entries in R is invertible if the first column is not the zero vector and if the second column is not in the span of the first column, which is to say, not a multiple of the first column. The same works for matrices with entries in Z/p (take this on faith). Your job is to find the number of elements in GL(2,Z/3). You first need to figure out the number of nonzero column vectors (two entries long) with entries in Z/3. Then, given a nonzero vector, you need to figure out how many vectors are not multiples of it. (c) If you succeed at (b), you probably can give me a formula for the number of elements of GL(2, Z/p) for any prime p. Try it!
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