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Unformatted text preview: CHAPTER 1: Vectors in R n Section 1.1: Vectors in R n For any positive integer n the set of all elements of the form ( x 1 , . . . , x n ) where x i ∈ R Definition R n for 1 ≤ i ≤ n is called n-dimensional Euclidean space and is denoted R n . The elements of R n are called points and are usually denoted P ( x 1 , . . . , x n ). 2-dimensional and 3-dimensional Euclidean space was originally studied by the ancient Greek mathematicians. In Euclid’s Elements, Euclid defines two and three dimensional space with a set of postulates, definitions and common notions. However, in modern mathematics, we not only want to make the concepts in Euclidean space more mathe- matically precise, but we want to be able to easily generalize the concepts to allow us to use the same ideas to solve problems in other areas. In Linear Algebra, we will view R n as a set of vectors rather than as a set of points. In particular, we will write an element of R n as a column vector and denote it with the usual vector symbol vectorx (Note that some textbooks, like the one we’re using, will denote vectors in bold face i.e. x ). That is, vectorx ∈ R n can be represented as vectorx = x 1 . . . x n , x i ∈ R , 1 ≤ i ≤ n. For example, in 3-dimensional Euclidean space, we will denote the origin (0 , , 0), by the vector vector 0 = . Imporatant Remarks: 1. Although we are going to view R n as a set of vectors, it is important to understand that we really are still referring to n-dimensional Euclidean space. In some cases it will be useful to think of the vector x 1 . . . x n as the point ( x 1 , . . . , x n ). In particular, we will sometimes interpret a set of vectors as a set of points to get a geometric object (such as a line or plane). 2. In Linear Algebra all vectors in R n should be written as column vectors, however, we will sometimes write these as n-tuples to match the notation that is commonly used in other areas. In particular, when we are dealing with functions of vectors, we will write f ( x 1 , . . ., x n ) rather than f x 1 . . . x n . 2 Chapter 1 Vectors in R n One advantage in viewing the elements of R n as vectors instead of as points is that we can perform operations on vectors. You likely have seen vectors in R 2 and R 3 in Physics being used to represent motion or force. From these physical examples we can observe that we add vectors by summing their components and multiply a vector by a scalar by multiplying each entry of the vector by the scalar. We keep this definition for vectors in R n . Let vectorx = x 1 . . . x n and vector y = y 1 ....
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