Matrices - ME 210 Applied Mathematics for Mechanical...

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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 INTRODUCTION TO LINEAR ALGEBRA Matrices and Vectors Matrix: A rectangular array of scalars (numbers, variables, or functions, real or complex). [ ] [ ] A a a a a a a a a a a ij mn 2 m 1 m n 2 22 21 n 1 12 11 = = Elements Rows Columns Element a i,j i th row j th column Size or dimension of a matrix: m x n Total no of rows Total no of columns
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Vector: A matrix with only one row (size 1×n, row vector ) or only one column (size m×1, column vector) . Equality of matrices: Two matrices [A] and [B] are said to be equal to each other if and only if i) they have the same dimension m×n, and ii) their corresponding elements are equal; i.e., a ij = b ij for all i = 1, 2, …, m and j = 1, 2, …, n Addition/Subtraction of matrices: Addition/subtraction is defined only for matrices of the same size and result in another matrix of the same size. For two matrices [A] and [B] of the same size [C] = [A] ± [B] implies that c ij = a ij ± b ij for all i = 1, 2, …, m and j = 1, 2, …, n
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Example: Given = = 0 1 3 0 1 5 ] B [ and 2 1 0 3 6 4 ] A [ = + + + + + = + = + 2 2 3 3 5 1 0 2 1 1 3 0 0 3 1 6 5 4 0 1 3 0 1 5 2 1 0 3 6 4 ] B [ ] A [ Multiplication/Division by a scalar: Multiplication/division of a matrix by a scalar k implies that all its elements are to be multiplied/divided by the same scalar k; i.e., k[A] = [ka ij ] or [A]/k = [a ij /k]
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Example: Given = 6 . 3 9 . 0 8 . 1 7 . 2 ] A [ = = 4 1 2 3 ] A [ 9 10 and 2 . 7 8 . 1 6 . 3 4 . 5 ] A [ 2 Then Some important properties of matrices: Given matrices [A], [B], and [C] of the same size and a set of scalar constants k 1 , k 2 , and k 3 , the following properties hold: [A] + [B] = [B] + [A] commutative [A] + ([B] + [C]) = ([A] + [B]) + [C]) associative k 1 ([A] + [B]) = k 1 [A] + k 1 [B] distributive (k 1 + k 2 ) [A] = k 1 [A] + k 2 [A] k 1 (k 2 [A]) = k 2 (k 1 [A]) = (k 1 k 2 ) [A]
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010
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This note was uploaded on 07/31/2011 for the course ME 210 taught by Professor Farukarinç during the Spring '09 term at Middle East Technical University.

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Matrices - ME 210 Applied Mathematics for Mechanical...

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