ALA240-2009 - November 19th

ALA240-2009 - November 19th - November 19th 2009 MAT240...

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November 19 th , 2009 MAT240: Abstract Linear Algebra Lecture: Suppose 𝐴 ∈ 𝑀 ??? can be row-reduced to I. Find 𝐴 1 . ? 𝑘 … ? 2 ? 1 𝐴 = 𝐼 → 𝐴 1 = ? 𝑘 … ? 2 ? 1 Turned practical the wrong way: i. Row reduce A to I, write all elementary matrices you ve encountered. ii. Find 𝐴 1 by multiplying these elementary matrices in the opposite order. 𝐴 1 = 𝐴 1 𝐼 = ? 𝑘 ( ? 2 ? 1 𝐼 ) Moral: 𝐴 1 is the result of doing to I the same row operations you did to A, to get to I. Efficient Way: 𝐴 | 𝐼 ?? ??? ??𝑒???𝑖??? 𝐼 | 𝐴 1 Eg. 1 0 0 1 | 1 0 0 1 Eg. 1 2 3 4 | 1 0 0 1 𝑅 2 3 𝑅 1 1 2 0 2 | 1 0 3 1 1 2 𝑅 2 1 2 0 1 | 1 0 3 2 1 2 𝑅 1 2 𝑅 2 1 0 0 1 | 2 0 3 2 1 2 1 2 3 4 1 = 2 0 3 2 1 2 How far can we go with row reduction? Reduced Row Echelon Form: 1. Make the first non-zero entry in any row be 1 (the pivot). 2. In the column of every pivot, all else is 0.
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