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notes_2 - Notes For Optimization Theory & Analysis (ESE304)...

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Notes For Optimization Theory & Analysis (ESE304) Michael A. Carchidi September 14, 2010 Chapter 2 - The Algebra of Scalars, Matrices and Vectors The following notes are based on the text entitled: Operations Research by Wayne L. Winston (4th edition), and these can be viewed at https://courseweb.library.upenn.edu/ after you log in using your PennKey user name and Password. This chapter is provided as a review of the algebra of scalars, matrices and vectors. It should be read by those students who require this brief review. There are very few examples provided in this chapter of the notes. The student should refer to Chapter 2 of the text for more examples, if needed. 2.1 Some De f nitions A scalar is a real number and the algebra of scalars is the same algebra that you learned in high school. An m × n matrix A is an array of elements having m rows and n columns. The size of A is said to be m × n (read m n ), and the elements of A are scalars. The element of A in the i th row and j th column
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is represented by the scalar a ij . All of this information is summarized by writing A = a 11 a 12 a 13 ··· a 1 j a 1 n a 21 a 22 a 23 a 2 j a 2 n a 31 a 32 a 33 a 3 j a 3 n . . . . . . . . . . . . . . . . . . . . . a i 1 a i 2 a i 3 a ij a in . . . . . . . . . . . . . . . . . . . . . a m 1 a m 2 a m 3 a mj a mn =[ a ij ] m × n where each a ij is a real number (i.e., scalar). A matrix having only one column is called a column vector and is represented as u = u 1 u 2 u 3 . . . u i . . . u m u i ] m × 1 where each u i is a scalar. A matrix having only one row is called a row vector and is represented as v = h v 1 v 2 v 3 v j v n i v j ] 1 × n , where each v j is a scalar. In these notes we shall represent scalars using italic type. We shall represent matrices using UPPERCASE bold type and we shall represent vectors using lowercase bold type. 2.2 Operations on Matrices Equality : Two matrices A and B areequa liftheyhavethesames izeandthe same corresponding elements. In other words, if A a ij ] m × n and B b ij ] m × n , then A = B ,i fandon lyi f a ij = b ij for all i =1 , 2 , 3 ,...,m and j , 2 , 3 ,...,n . Note that only matrices of the same size can be compared in this way. Example : Solve for x , y and z if " 12 x 3 4 x + y 6 # = " 1 y 2 xy x 4 z 6 # . 2
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In order for these two matrices to be equal, we must have 2 x = y 2 x, 3= y x and x + y = z which leads to x =1 , y =4 ,and z =5 . Addition :I f A and B havethesames izeand A =[ a ij ] m × n and B b ij ] m × n , then C = A + B c ij ] m × n provided that c ij = a ij + b ij for all i , 2 , 3 ,...,m and j , 2 , 3 ,...,n . In other words, [ a ij ] m × n +[ b ij ] m × n a ij + b ij ] m × n . In words, when you add two matrices of the same size, you add corresponding elements. Note that only matrices of the same size can be added. Example : " 123 456 # + " 789 10 11 12 # = " 1+7 2+8 3+9 4+10 5+11 6+12 # = " 81 01 2 14 16 18 # .
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This note was uploaded on 03/02/2011 for the course ESE 304 taught by Professor Michaela.carchidi during the Winter '11 term at UPenn.

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notes_2 - Notes For Optimization Theory & Analysis (ESE304)...

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