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SJSU ISE 230 - Basic Linear Algebra

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Chapter 2 Basic Linear Algebra to accompany Operations Research Applications and Algorithms 4th edition by Wayne L Winston Copyright c 2004 Brooks Cole a division of Thomson Learning Inc 2 1 Matrices and Vectors A matrix is any rectangular array of numbers 1 2 3 4 1 2 3 4 5 6 1 2 2 1 If a matrix A has m rows and n columns it is referred to as an m x n matrix m x n is the order of the matrix It is typically written as a 11 a 12 a 1n a 21 a 22 a 2n A a a a mn m1 m2 2 The number in the ith row and jth column of A is called the ijth element of A and is written aij Two matrices A aij and B bij are equal if and only if A and B are the same order and for all i and j aij bij If 1 2 A 3 4 and x y B w z A B if and only if x 1 y 2 w 3 and z 4 3 Any matrix with only one column is a column vector The number of rows in a column vector is the dimension of the column vector 1 2 Rm will denote the set all m dimensional column vectors Any matrix with only one row a 1 x n matrix is a row vector The dimension of a row vector is the number of columns 1 2 3 4 Any m dimensional vector either row or column in which all the elements equal zero is called a zero vector written 0 0 0 0 0 0 Any m dimensional vector corresponds to a directed line segment in the m dimensional plane For example the two dimensional vector u corresponds to the line segment joining the point 0 0 to the point 1 2 5 X2 1 2 X1 1 2 1 3 The directed line segments vectors u v w are shown u 1 2 v 1 3 w 1 2 6 The scalar product is the result of multiplying two vectors where one vector is a column vector and the other is a row vector For the scalar product to be defined the dimensions of both vectors must be the same u v u 1 v 1 u 2 v 2 u n v n The scalar product of u and v is written 2 v 1 u 1 2 3 2 u v 1 2 2 1 3 2 10 7 The Scalar Multiple of a Matrix Given any matrix A and any number c the matrix cA is obtained from the matrix A by multiplying each element of A by c 1 2 3 6 A 3 A 1 0 3 0 Addition of Two Matrices Let A aij and B bij be two matrixes with the same order Then the matrix C A B is defined to be the m x n matrix whose ijth element is aij bij Thus to obtain the sum of two matrixes A and B we add the corresponding elements of A and B 1 2 3 A 0 1 1 1 2 3 B 2 1 1 0 1 1 2 2 3 3 0 0 C 0 2 1 1 1 1 2 0 2 8 This rule for matrix addition may be used to add vectors of the same dimension Vectors may be added using the parallelogram law or by using matrix addition v 2 1 u 1 2 u v 3 3 X2 3 3 3 1 2 2 u v u 1 2 1 v X1 1 2 3 9 Line segments can be defined using scalar multiplication and the addition of matrices If u 1 2 and v 2 1 the line segment joining u and v called uv is the set of all points in the mdimensional plane corresponding to the vectors cu 1 c v where 0 c 1 X2 c 1 2 c 1 2 u 1 c 0 v X1 1 2 10 The Transpose of a Matrix Given any m x n matrix the transpose of A written AT is the n x m matrix For any matrix A AT T A 11 Matrix Multiplication Given to matrices A and B the matrix product of A and B written AB is defined if and only if the number of columns in A the number of rows in B The matrix product C AB of A and B is the m x n matrix C whose ijth element is determined as follows ijth element of C scalar product of row i of A x column j of B 12 Example 1 Matrix Multiplication Computer C AB for A 1 1 2 2 1 3 1 1 B 2 3 1 2 Solution Because A is a 2x3 matrix and B is a 3x2 matrix AB is defined and C will be a 2x2 matrix C 1 1 1 2 2 5 1 C 1 2 1 3 2 7 1 11 21 C 1 1 1 2 3 8 2 C 1 2 1 3 3 11 2 12 22 5 8 C 7 11 13 Many computations that commonly occur in operations research can be concisely expressed by using matrix multiplication Some important properties of matrix multiplications are Row i of AB row i of A B 1 1 1 1 2 A B 2 3 2 1 3 1 2 Column j of AB A column j of B 14 Use the EXCEL MMULT function to multiply the matrices Enter matrix A into cells B1 D2 and matrix B into cells B4 C6 Select the output range B8 C9 into which the product will be computed In the upper left hand corner B8 of this selected output range type the formula A B C MMULT B1 D2 B4 C6 1 Matrix A 1 1 Press Control Shift Enter 2 3 4 5 6 7 8 9 2 1 Matrix B 1 2 1 1 3 2 AB 1 7 2 11 D 2 3 15 2 2 Matrices and Systems of Linear Equations Consider a system of linear equations a11 x1 a12 x2 a1n xn b1 a21 x1 a22 x2 a2n xn b1 am1 x1 am2 x2 amn xn bm The variables or unknowns are referred to as x1 x2 xn while the aij s and bj s are constants A set of such equations is called a linear system of m equations in n variables A solution to a linear set of m equations in n unknowns is a set of values for the unknowns that satisfies each of the system s m equations 16 Example 5 Solution to Linear System Show that 1 x 2 a solution to the linear system x1 2x2 and that 5 2x 1 x 2 0 3 x 1 is not a solution to the linear system 17 Example 5 Solution To show that 1 x 2 is a solution x1 1 and x2 2 must be substituted in both equations The equations must be satisfied …


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SJSU ISE 230 - Basic Linear Algebra

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