(BQ) Part 2 book "Mathematics and statistics for financial risk management" has contents: Vector spaces, linear regression analysis, time series models, decay factors. | chapter 9 Vector Spaces I n this chapter we introduce the concept of vector spaces. At the end of the chapter we introduce principal component analysis and explore its application to risk management. Vectors Revisited In the previous chapter we stated that matrices with a single column could be referred to as vectors. While not necessary, it is often convenient to represent vectors graphically. For example, the elements of a 2 × 1 matrix can be thought of as representing a point or a vector in two dimensions,1 as shown in Exhibit . v1 = 10 () 2 Similarly, a 3 × 1 matrix can be thought of as representing a point or vector in three dimensions, as shown in Exhibit . 5 v 2 = 10 () 4 While it is difficult to visualize a point in higher dimensions, we can still speak of an n × 1 vector as representing a point or vector in n dimensions, for any positive value of n. In addition to the operations of addition and scalar multiplication that we explored in the previous chapter, with vectors we can also compute the Euclidean inner product, often simply referred to as the inner product. For two vectors, the Euclidean 1 In physics, a vector has both magnitude and direction. In a graph, a vector is represented by an arrow connecting two points, the direction indicated by the head of the arrow. In risk management, we are unlikely to encounter problems where this concept of direction has any real physical meaning. Still, the concept of a vector can be useful when working through the problems. For our purposes, whether we imagine a collection of data to represent a point or a vector, the math will be the same. 169 170 Mathematics and Statistics for Financial Risk Management 10 5 0 –10 –5 0 5 10 –5 –10 Exhibit Two-Dimensional Vector z x y Exhibit Three-Dimensional Vector 171 Vector Spaces inner product is defined as the sum of the product of the corresponding elements in the vector. For two vectors, a and b, we denote the
