Lect4 - 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 Denote A = a 1 1 a 1 2 ··· a 1 n a 2 1 a 2 2 ··· a 2 n . . . . . . . . . . . . a m 1 a m 2 ··· a m n , x = x 1 x 2 . . . x n , b = b 1 b 2 . . . b n and a j = j th column of A . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Linear Combination Denote A = a 1 1 a 1 2 ··· a 1 n a 2 1 a 2 2 ··· a 2 n . . . . . . . . . . . . a m 1 a m 2 ··· a m n , x = x 1 x 2 . . . x n , b = b 1 b 2 . . . b n and a j = j th column of A . The system A x = b can also be expressed as x 1 a 1 + x 2 a 2 + ··· + x n a n = b . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Linear Combination Denote A = a 1 1 a 1 2 ··· a 1 n a 2 1 a 2 2 ··· a 2 n . . . . . . . . . . . . a m 1 a m 2 ··· a m n , x = x 1 x 2 . . . x n , b = b 1 b 2 . . . b n and a j = j th column of A . The system A x = b can also be expressed as x 1 a 1 + x 2 a 2 + ··· + x n a n = b . L.H.S. is called a linear combination of a 1 , a 2 , ··· , a n . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Linear Combination - Geometric Meaning Example . For what a 1 , a 2 and b will the system 2 x 1 + 3 x 2 = 8 5 x 1- 4 x 2 = 7 be expressed in the form of x 1 a 1 + x 2 a 2 = b ? Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Linear Combination - Geometric Meaning Example . For what a 1 , a 2 and b will the system 2 x 1 + 3 x 2 = 8 5 x 1- 4 x 2 = 7 be expressed in the form of x 1 a 1 + x 2 a 2 = b ? Geometric Interpretation Solving the system passes to expressing b as a sum of a 1 and a 2 after “suitable” scaling. 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 EquationsChapter 1....
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Lect4 - Chapter 1. Matrices and Systems of Equations...

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