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# a1 - CO350 Assignment 1 Alex Fok 20300650 Instructor...

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CO350 Assignment 1 Alex Fok 20300650 , May 14 2010 : - Instructor Mathieu Guay Paquet ( ) 1 a Given = , Ax b has a solution s Columns of A are not linearly independent Proof = Let K be the solution set of the system Ax b = Let KH be the solution set of the corresponding homogenous system Ax 0 . Let LA denote the left multiplication transformation - Let ai denote the i th column of A , Since the columns of A are not linearly independent = ,…, > rankA dimspana1 an n = ( ). , Note that KH N LA By dimension theorem = - = - > - = dimKH n rankLA n rankA n n 0 , - . Therefore there exists a non trivial solution u KH + , Consider s tu for t R + = + = = As tu As tAu As b , + . , . Hence s tu K That is there are infinitely many solutions ( ) 1 b Given = , Ax b has a solution s Columns of A are linearly independent Proof ( ). Using the same notation in part a , Since the columns of A are linearly independent = ,…, = rankA dimspana1 an n = ( ). , Note that KH N LA By dimension theorem = - = - = - = dimKH n rankLA n rankA n n 0 , . = + ={ }. Therefore KH is trivial K s KH s , = . Thus Ax b has a unique solution 2 ( : = , ≥ ) , Suppose max ctx Ax b x 0

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a1 - CO350 Assignment 1 Alex Fok 20300650 Instructor...

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