Linear+algebra+review.pdf - ISyE6669 Review on Linear...

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ISyE6669 Review on Linear Algebra and Basic Concepts of Convexity Fall 2017 Andy Sun In this lecture, we give a review of some basic concepts in linear algebra. We also introduce the concept of convexity, including convex combination, convex hull, and convex sets. These are very important objects that we will study deeper in this class. Notation 1. Vector: x = x 1 . . . x n R n is a vector in the Euclidean space R n . In this class, we will denote vectors in bold lower case letters, and matrices in bold upper case letters. A standard unit vector in R n is defined as the vector with one element equal to 1 and all other elements equal to 0. There are n standard unit vectors in R n : e 1 = 1 0 . . . 0 , e 2 = 0 1 0 . . . 0 , . . . , e n = 0 . . . 0 1 . The transpose of a vector x is denoted as x > . 2. Inner product of two vectors: x > y = y > x = n i =1 x i y i . 3. Matrix: A R m × n is an m -by- n matrix with entries of real numbers: A = a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a m 1 a m 2 . . . a mn = A 1 A 2 . . . A n = a > 1 a > 2 . . . a > m where A i = [ a 1 i , a 2 i , . . . , a mi ] > is the i -th column of A , and a > i is the i -th row of A . Why do we write a matrix in both the column form and the row form, although they are obviously equivalent? Because having at least these two ways of viewing a matrix is very helpful. For example, matrix-vector multiplication can be written in two ways. 1
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4. Matrix-vector multiplication: Ax = A 1 A 2 . . . A n x 1 . . . x n = n X i =1 A i x i = a > 1 a > 2 . . . a > m x = a > 1 x a > 2 x . . . a > m x 1 Linear, Affine, and Convex Combinations Definition 1 (Linear, Affine, Convex Combination) . Let x 1 , ..., x k be k points in R n . Then 1. A point x R n is called a linear combination of x 1 , ..., x k if y can be expressed as y = k X i =1 λ i x i where λ i R for all i = 1 , ..., k .
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