Linear Algebra 2011

Linear Algebra 2011 - 3/22/2011 LINEAR ALGEBRA REVIEW Math...

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3/22/2011 Math Modeling, I (GAUSS project) 1 LINEAR ALGEBRA REVIEW Math Modeling I, Spring 2011 Linear Algebra View a matrix as an “array of columns”: A mxn = [ a 1 , a 2 ,…, a n ] with a i =[a 1i ,a 2i ,…,a mi ] T If x=[x 1 ,x 2 ,…,x n ] T , then Ax can be viewed as a linear combination of the columns of A: Ax = x 1 a 1 +x 2 a 2 +…+x n a n A matrix is symmetric if A=A T A matrix is orthogonal if its columns are orthogonal vectors: a i T a j =0 for all i j; a i T a i =1. Or, (to be proved on next slide) A T A=I.
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3/22/2011 Math Modeling, I (GAUSS project) 2 Important Theorems Theorem 1: If A mxn is an orthogonal matrix (m n), then A T A=I nxn Proof. Let A = [ a 1 , a 2 ,…, a n ]. Since A is orthogonal, a i T a j =0 for all i j; a i T a i =1 Now, the (i,j) entry of A T A= a i T a j = δ ij. So, A T A=I. Theorem 2. A T A and AA T are symmetric matrices. Important Theorems, cont’d
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Linear Algebra 2011 - 3/22/2011 LINEAR ALGEBRA REVIEW Math...

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