Lect6 - Chapter 1. Matrices and Systems of Equations...

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Unformatted text preview: Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Linear Combination - Definition Definition Let a 1 , a 2 , ··· , a n be vectors in R n , and c 1 , c 2 , ··· , c n be scalars. Then a sum of the form c 1 a 1 + c 2 a 2 + ··· + c n a n is called a linear combination of the vectors a 1 , a 2 , ··· , a n . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Linear Combination - Definition Definition Let a 1 , a 2 , ··· , a n be vectors in R n , and c 1 , c 2 , ··· , c n be scalars. Then a sum of the form c 1 a 1 + c 2 a 2 + ··· + c n a n is called a linear combination of the vectors a 1 , a 2 , ··· , a n . Example . Is    1 2    a linear combination of    1 1    ,    2 1    ,    1 3    ? Is       1 1 2       a linear combination of       1 1 1       ,       1       ,       1 1       ? Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Consistency Theorem A x = b is consistent if and only if b is a linear combination of column vectors of A . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Consistency Theorem A x = b is consistent if and only if b is a linear combination of column vectors of A . Proof. “if”: Write A = ( a 1 a 2 ··· a n ) . Suppose b is a linear combination of a 1 , a 2 , ··· , a n . It means that we have some scalars α 1 , α 2 , ··· , α n such that α 1 a 1 + α 2 a 2 + ··· + α n a n = b . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Consistency Theorem A x = b is consistent if and only if b is a linear combination of column vectors of A . Proof. “if”: Write A = ( a 1 a 2 ··· a n ) . Suppose b is a linear combination of a 1 , a 2 , ··· , a n . It means that we have some scalars α 1 , α 2 , ··· , α n such that α 1 a 1 + α 2 a 2 + ··· + α n a n = b . Take x =          α 1 α 2 . . . α n          , then A x = b . i.e. The equation A x = b has solution. Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Consistency (Cont’d) Proof. “only if”: Let          a 1 a 2 ....
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This note was uploaded on 04/30/2010 for the course MATH 1111 taught by Professor Dr,li during the Spring '10 term at HKU.

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Lect6 - Chapter 1. Matrices and Systems of Equations...

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