Transcription of Vectors and Matrices A - MIT
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Vectors and Matrices A - MIT
Vectors and Matrices Appendix A. Vectors and Matrices are notational conveniences for dealing with systems of linear equations and inequalities. In particular, they are useful for compactly representing and discussing the linear programming problem: n X. Maximize cjxj, j=1. subject to: n X. ai j x j = bi (i = 1, 2, .. , m), j=1. xj 0 ( j = 1, 2, .. , n). This appendix reviews several properties of Vectors and Matrices that are especially relevant to this problem. We should note, however, that the material contained here is more technical than is required for understanding the rest of this book. It is included for completeness rather than for background. Vectors . We begin by defining Vectors , relations among Vectors , and elementary vector operations. Definition. A k-dimensional vector y is an ordered collection of k real numbers y1 , y2 , .. , yk , and is written as y = (y1 , y2 , .. , yk ). The numbers y j ( j = 1, 2, .. , k) are called the components of the vector y. Each of the following are examples of Vectors : i) (1, 3, 0, 5) is a four-dimensional vector.
will later prove a number of properties of vectors that do not have straightforward generalizations to matrices. Definition. A k-by-1 matrix is called acolumn vector and a 1-by-k matrix is called a row vector. Thecoefficientsinrow i ofthematrix Adeterminearowvector Ai =(ai1,ai2,…, ain),andthecoefficients
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