O'Reilly logo

Mathematics and Statistics for Financial Risk Management by Michael B. Miller

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

CHAPTER 7

Vector Spaces

In 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 Figure 7.1.

(7.1) Numbered Display Equation

Similarly, a 3 × 1 matrix can be thought of as representing a point or vector in three dimensions, as shown in Figure 7.2.

(7.2) Numbered Display Equation

FIGURE 7.1 Two-Dimensional Vector

nc07f001.eps

FIGURE 7.2 Three-Dimensional Vector

nc07f002.eps

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, ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required