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ALA240-2009 - November 17th

# ALA240-2009 - November 17th - Page 1 of 4 November 17 th...

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Unformatted text preview: Page 1 of 4 November 17 th , 2009 MAT240: Abstract Linear Algebra Lecture: Question 1: Can we choose P, Q so that PAQ is “simpler”? Question 2: What’s “simpler”? Answer to Question 2: Claim: Let ? = ? ¡ then ???¢?£ = ? , where k is the number of row/columns of I ? : ¤ ? → ¤ ? , ¥ → ?¥ ¢£ = ¢ £ = ⁡ ( ¢ £ ) ¢ ? £ = ¦??§ 1 … ? ¨ ???¢ ? £ = ? “Elementary Matrices” Three kinds © ª , « 1 , © ª , 2 , © ª , « , 3 1. © ª , « 1 = interchange row/column i with row/column j © ª , « 1 ¬ = ¬ with rows i,j interchanged. ¬© ª , « 1 = ¬ with columns i,j interchanged. Claim: © ª , « 1 is invertible and ­© ª , « 1 ® − 1 = © ª , « 1 Proof: © ª , « 1 © ª , « 1 = 2. © ª , 2 = multiply row or column I by a constant c (c cannot equal 0). If c = 0, then the matrix is not invertible....
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ALA240-2009 - November 17th - Page 1 of 4 November 17 th...

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