Science:Math Exam Resources/Courses/MATH307/April 2006/Question 01 (a)
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Question 01 (a) 

Let PA = LU be and

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Hint 

Science:Math Exam Resources/Courses/MATH307/April 2006/Question 01 (a)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. This matrix A in this question is given in the form of an LU decomposition with partial pivoting where P is just the matrix which swaps the rows of A. The matrix P also would not affect R(A), N(A), R(A^{T}) where we need to look at the matrix U (echelon form) to find the bases. To find a basis for the R(A), we must identify the pivot columns in matrix U, which turns out to be the first and second column. The pivots are highlighted in red. The basis for R(A) is then the corresponding columns in A to the pivot columns in U. The pivot columns of A are highlighted in red.
And therefore
From row reducing A^{T}, you should acquire the matrix Solving for x, you end up with 2 equations and 1 free variable. And hence 