1. Linear Algebra Chapter 4 n Linear Algebra with Applications –-Gareth Williams n Br. Joel Baumeyer, F.S.C.

2. Vector Spaces n Definition: Let [u 1, u 2, … u n,] be a sequence of real numbers. The set of all such sequences is called n-space and is denoted by  n.  n = {[u 1, u 2, … u n,] : u i  , i  {1, 2, …, n}

3. Vector Spaces (continued) n Zero vector: 0 = [0,0, …, 0] n -v = -1v

4. Vector Spaces (continued)

5. Dot Product

6. Vector Properties

7. Cauchy-Schwartz Inequality

8. Orthogonal Vectors n Two vectors are orthogonal by definition if the angle between them is a right angle.

10. Norm and Distance n Def: Let x = [x 1, x 2,…, x n ] & y = [y 1, y 2,…, y n ] be two vectors in  n. The distance between x and y is de- noted by d(x,y)  ||x - y|| and is defined by

11. Geometrical Structure of  n