hw4_solu

# hw4_solu - Solutions for Selected problems of Homework 4 A...

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Unformatted text preview: Solutions for Selected problems of Homework 4 A. Agarwal 1016-331 Linear Algebra-1 Spring 2009-10 1. Q24,P.151 Here is the product AB as a combination of rows of B AB = 1 parenleftBig 2 3 0 parenrightBig + parenleftBig 1 − 1 1 parenrightBig + − 2 parenleftBig − 1 6 4 parenrightBig − 3 parenleftBig 2 3 0 parenrightBig + 1 parenleftBig 1 − 1 1 parenrightBig + 1 parenleftBig − 1 6 4 parenrightBig 2 parenleftBig 2 3 0 parenrightBig + parenleftBig 1 − 1 1 parenrightBig + − 1 parenleftBig − 1 6 4 parenrightBig = 4 − 9 − 8 − 6 − 4 5 5 − 4 Note that the multiplier red entries are row entries from A . 2. Q30, P.151 We shall see that the lin ear com bi na tions of the rows of B approach that we used to compute AB in the previous problem will be useful here as well. Let the matrix A be written in its row form, A n × k = arrowaxisleftaxisshortaxisshort a 1 axisshortaxisshortarrowaxisright arrowaxisleftaxisshortaxisshort a 2 axisshortaxisshortarrowaxisright . . . arrowaxisleftaxisshortaxisshort a n axisshortaxisshortarrowaxisright then, A n × k B k × p = arrowaxisleftaxisshortaxisshort a 1 axisshortaxisshortarrowaxisright arrowaxisleftaxisshortaxisshort a 2 axisshortaxisshortarrowaxisright . . . arrowaxisleftaxisshortaxisshort a n axisshortaxisshortarrowaxisright B = arrowaxisleftaxisshortaxisshort a 1 B axisshortaxisshortarrowaxisright arrowaxisleftaxisshortaxisshort a 2 B axisshortaxisshortarrowaxisright . . . arrowaxisleftaxisshortaxisshort a n B axisshortaxisshortarrowaxisright Since we are given that the rows of A are linearly dependent , this means the following equation has a non-zero solution (at least one c i negationslash = 0) c 1 a 1 + c 2 a 2 + ... + c n a n = (1) If we right- mul ti ply by ma trix B on both sides of equation (1), we get c 1 a 1 B + c 2 a 2 B + ... + c n a n B = (2) 1 Since a i B is just the i − th row of AB , this implies the rows of AB are satisfying equation (2) with a non-zero solution. (remember at least onesolution....
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hw4_solu - Solutions for Selected problems of Homework 4 A...

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