Arithmetic Groups and Their Generalizations What, Why, and - download pdf or read online

By Lizhen Ji

ISBN-10: 0821846752

ISBN-13: 9780821846759

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27 Let A =  1 0 3 1 Set up the augmented matrix (A|I)  1 2   1 0 3 1  1 0 0  0 1 0  0 0 1 2 2 −1 Next take (−1) times the first row and add to the row added to the last. This yields  1 2 2 1  0 −2 0 −1  0 −5 −7 −3 Then take 5 times the second row and add to  1 2 2   0 −10 0 0 0 14 second followed by (−3) times the first  0 0  1 0 . 0 1 −2 times the last row.  1 0 0  −5 5 0  1 5 −2 Next take the last row and add to (−7) times the top row. This yields   −7 −14 0 −6 5 −2    0 −10 0 −5 5 0  .

20) is simplified to a·a ≥ 0. The Cauchy Schwarz inequality allows a proof of the triangle inequality for distances in Fn in much the same way as the triangle inequality for the absolute value. 25) and equality holds if and only if one of the vectors is a nonnegative scalar multiple of the other. 26) ||a| − |b|| ≤ |a − b| Proof : By properties of the inner product and the Cauchy Schwarz inequality, 2 |a + b| = (a + b) · (a + b) = (a · a) + (a · b) + (b · a) + (b · b) 2 2 2 = |a| + 2 Re (a · b) + |b| ≤ |a| + 2 |a · b| + |b| 2 2 2 2 ≤ |a| + 2 |a| |b| + |b| = (|a| + |b|) .

3 Let A = Aij be an m × n matrix and let v be an n × 1 matrix,   v1  .  .  v =  .  , A = (a1 , · · · , an ) vn where ai is an m × 1 vector. Then Av, written as  ( ) a1 · · · an    v1 ..   . , vn is the m × 1 column vector which equals the following linear combination of the columns. 9) 42 CHAPTER 2. LINEAR TRANSFORMATIONS If the j th column of A is       A1j A2j .. 9) takes the form    v1    A11 A21 .. Am1        + v2      A12 A22 .. Am2        + · · · + vn      A1n A2n ..

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Arithmetic Groups and Their Generalizations What, Why, and How by Lizhen Ji


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