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

MATH307 December 2010

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Question 02 (d)
Suppose that $A={\begin{bmatrix}1&1&0&1&0\\2&2&1&3&0\\3&3&1&4&1\\4&4&1&5&1\end{bmatrix}}\quad {\text{rref}}(A)={\begin{bmatrix}1&1&0&1&0\\0&0&1&1&0\\0&0&0&0&1\\0&0&0&0&0\end{bmatrix}}$ (d) What are rank(A) and dim(N(AT))?

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Hint
The rank of A is the number of pivot columns in rref(A). Further, by the Rank-nullity theorem, for a m×n matrix A it holds that rank(A) + dim(N(A)) = n. What does that imply for AT?

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Solution
Found a typo? Is this solution unclear? Let us know here. 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 Rank-nullity theorem, since AT is a 5×4 matrix: dim(N(AT)) = 4 - rank(AT). Since the rank of A equals the rank of AT we obtain dim(N(AT)) = 4 - 3 = 1.

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