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Unformatted text preview: Differential Equations & Linear Algebra Lecture 13 Jianli XIE Email: [email protected] Department of Mathematics, SJTU Fall 2010 Contents Elementary linear algebra Properties of matrices Contents Elementary linear algebra Properties of matrices System of linear equations Contents Elementary linear algebra Properties of matrices System of linear equations Rowechelon form Contents Elementary linear algebra Properties of matrices System of linear equations Rowechelon form Solving systems of linear algebraic equations The GaussJordan algorithm Introduction to matrices A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns, denoted by A = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . . . . . . . a m 1 a m 2 ··· a mn . (1) We speak of A as an m × n matrix. Matrices will usually be denoted by capital letters. The element (entry) lying in the i th row and j th column is designated by a ij , the first subscript identifying its row and the second its column. Equality of matrices, zero matrix Sometimes the notation ( a ij ) is used to denote the matrix whose generic element is a ij . It is also convenient to write a ij = ( A ) ij . The matrices A and B are said to be equal if A and B have the same size and corresponding elements are equal. The zero matrix of size m × n , denoted by the symbol , is the matrix whose elements are all zero. Addition The sum of two m × n matrices A and B is defined as the matrix obtained by adding corresponding elements: A + B = ( a ij ) + ( b ij ) = ( a ij + b ij ) . (2) With this definition, it follows that matrix addition is commutative and associative, so that A + B = B + A , A +( B + C ) = ( A + B )+ C . (3) Scalar multiple, subtraction The product of a matrix A by a number (scalar) α is defined as follows: α A = α ( a ij ) = ( αa ij ) . (4) The distributive laws α ( A + B ) = α A + α B , ( α + β ) A = α A + β A (5) are satisfied. In particular, the negative of A , denoted by A , is defined by A = ( 1) A . Then the difference A B of two m × n matrices is defined by A B = A + ( B ) . Thus A B = ( a ij ) ( b ij ) = ( a ij b ij ) . (6) Multiplication The product AB of two matrices is defined whenever the number of columns in the first factor, A , is the same as the number of rows in the second factor, B . If A and B are m × n and n × r matrices, respectively, then the product C = AB is an m × r matrix. The element in the i th row and j th column of C is found by multiplying each element of the i th row of A by the corresponding element of the j th column of B and then adding the resulting products. In symbols, c ij = n X k =1 a ik b kj . (7) Multiplication By direct calculation, it can be shown that matrix multiplication satisfies the associative law ( AB ) C = A ( BC ) (8) and the distributive law A ( B + C ) = AB + AC . (9) However, in general, matrix multiplication is not commutative. For both products AB and BA...
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This note was uploaded on 05/17/2011 for the course ME 255 taught by Professor Jianlixie during the Fall '10 term at Shanghai Jiao Tong University.
 Fall '10
 JianliXie

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