Science:Math Exam Resources/Courses/MATH307/December 2010/Question 02 (d)
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Question 02 (d) 

Suppose that (d) What are rank(A) and dim(N(A^{T}))? 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

The rank of A is the number of pivot columns in rref(A). Further, by the Ranknullity theorem, for a m×n matrix A it holds that rank(A) + dim(N(A)) = n. What does that imply for A^{T}? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. First, rank(A) is the number of pivot columns in A (or rref(A)), which is 3. From the Ranknullity theorem, since A^{T} is a 5×4 matrix: dim(N(A^{T})) = 4  rank(A^{T}). Since the rank of A equals the rank of A^{T} we obtain dim(N(A^{T})) = 4  3 = 1. 