math125-lecture 2.2

math125-lecture 2.2 - A matrix is in reduced echelon form...

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Unformatted text preview: A matrix is in reduced echelon form if it is in row echelon form and has the additional property that above each leading 1 are 0's This is also called reduced row echelon form (rref) Sec 2.2 Friday, February 05, 2010 4:32 PM Section2dot2 Page 1 A matrix is in reduced echelon form if it is in row echelon form and has the additional property that above each leading 1 are 0's This is also called reduced row echelon form (rref) This is a process for eliminating the matrix entries above leading 1's i.e. reducing those entries to 0's Remark: For an augmented matrix, back addition is analogous to doing backward substitution inside the matrix 2.2.1 Friday, February 05, 2010 4:32 PM Section2dot2 Page 2 A matrix is in reduced echelon form if it is in row echelon form and has the additional property that above each leading 1 are 0's This is also called reduced row echelon form (rref) This is a process for eliminating the matrix entries above leading 1's i.e. reducing those entries to 0's Remark: For an augmented matrix, back addition is analogous to doing backward substitution inside the matrix 1. Move the matrix into row echelon form (i.e. perform Gaussian elimination) 2. Move from the lower right leading 1 to the upper left, using leading 1's to eliminate/annihilate entries above each leading 1 2.2.2 Friday, February 05, 2010 4:32 PM Section2dot2 Page 3 1 2 1 4 3 7 -2 1-2 3 3 -1 1 2 1 4 0 1 -5 -11 0 7 5 7 R' 2 = R 2- 3 R 1 R' 3 = R 3 + 2 R 1 2.2.3 Friday, February 05, 2010 4:36 PM Section2dot2 Page 4 1 2 1 4 3 7 -2 1-2 3 3 -1 1 2 1 4 0 1 -5 -11 0 7 5 7 1 2 1 4 0 1 -5 -11 0 0 40 84 R' 2 = R 2- 3 R 1 R' 3 = R 3 + 2 R 1 R' 3 = R 3-7 R 4 2.2.4 Friday, February 05, 2010 4:36 PM Section2dot2 Page 5 1 2 1 4 3 7 -2 1-2 3 3 -1 1 2 1 4 0 1 -5 -11 0 7 5 7 1 2 1 4 0 1 -5 -11 0 0 40 84 1 2 1 4 0 1 -5 -11 0 0 1 21/10 R' 2 = R 2- 3 R 1 R' 3 = R 3 + 2 R 1 R' 3 = R 3-7 R 4 R' 3 = 1/40 R 2 now in ref 2.2.5 Friday, February 05, 2010 4:36 PM Section2dot2 Page 6 1 2 1 4 3 7 -2 1-2 3 3 -1 1 2 1 4 0 1 -5 -11 0 7 5 7 1 2 1 4 0 1 -5 -11 0 0 40 84 1 2 1 4 0 1 -5 -11 0 0 1 21/10 1 2 0 19/10 0 1 0 -1/2 0 0 1 21/10 R' 2 = R 2- 3 R 1 R' 3 = R 3 + 2 R 1 R' 3 = R 3-7 R 4 R' 3 = 1/40 R 2 R' 1 = R 1 - 2 R 2 now in ref Begin back addition: R' 2 = R 2 + 5 R 3 R' 1 = R...
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math125-lecture 2.2 - A matrix is in reduced echelon form...

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