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Lecture 5 - Review of Linear Equations

# Lecture 5 - Review of Linear Equations - Review of linear...

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Lecture 5 Review of Linear Equations UCSD Math 171A: Numerical Optimization Philip E. Gill ( [email protected] ) Wednesday, January 14th, 2009 Review of linear equations We review properties of systems of linear equations Ax = b where A is an m × n matrix and b is an m -vector. We say A R m × n and b R m . We make no assumptions on the shape of A we cannot say that x = A - 1 b , in general. UCSD Center for Computational Mathematics Slide 2/35, Wednesday, January 14th, 2009 Example: finding the drug mixtures in a chemotherapy treatment Set of m typical patient types: patient 1 , patient 2 , . . . , patient m Set of n chemotherapy drugs: drug 1 , drug 2 , . . . , drug n x j = quantity of drug j in the medication a ij = estimated effect on patient i of drug j b i = desired effect of drug mixture on patient i UCSD Center for Computational Mathematics Slide 3/35, Wednesday, January 14th, 2009 Problem: Find x 1 , x 2 , . . . , x n , such that b i = a i 1 x 1 + a i 2 x 2 + · · · + a in x n for i = 1, 2, . . . , m This is the same as: b i = n X j =1 a ij x j for i = 1, 2, . . . , m UCSD Center for Computational Mathematics Slide 4/35, Wednesday, January 14th, 2009

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In matrix form: b = Ax , where A = a 11 a 12 · · · a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a m 1 a m 1 · · · a mn , b = b 1 b 2 . . . b m , x = x 1 x 2 . . . x n This is a system of linear equations. UCSD Center for Computational Mathematics Slide 5/35, Wednesday, January 14th, 2009 Another notation. Write A by columns: A = a 1 a 2 · · · a n with a j = a 1 j a 2 j . . . a mj In this case, a j = effects of drug j with a j R m , i.e., Ax = n X j =1 a j x j UCSD Center for Computational Mathematics Slide 6/35, Wednesday, January 14th, 2009 An obvious question: Is it possible to get the desired effect using this mixture? i.e., does there exist an x such that b = Ax ? UCSD Center for Computational Mathematics Slide 7/35, Wednesday, January 14th, 2009 Definition A system of equations is compatible if there exists a vector x such that b = Ax . i.e., there exist x 1 , x 2 , . . . , x n such that b = n X j =1 a j x j i.e., b can be written as a linear combination of the columns of A . A compatible system may be regarded as defining an expansion or representation of b in terms of the columns of A .
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Lecture 5 - Review of Linear Equations - Review of linear...

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