Unit 8
Introduction to Vector Spaces

Units 8 and 9 introduce you to the study of abstract vector spaces. How well you do in these units depends on how well you have learned the materials presented in Units 1-7. Two- and three-dimensional Euclidean space, which you studied in Units 6 and 7, provide the simplest examples of vector spaces, and it is easy to visualize examples in two and three dimensions. Solving problems or proving theorems in higher numbers of dimensions can be greatly simplified if you imagine what the situation in question would look like if it were reduced to three dimensions.

The mathematical machinery used in developing the theory of linear vector spaces is matrix algebra. This subject was presented in Units 1-5, and you should review these units now if you are uncertain of your command of matrix algebra.

The theory of linear vector spaces is used in many branches of mathematics. It plays a major role in statistics (e.g., in least-squares curve fitting, linear regression and factor analysis), and together with calculus, is the foundation of most mathematical work in the sciences. In physics, for example, all of quantum mechanics is based on a kind of vector space called a Hilbert space. In chemistry, biology, psychology and even sociology, linear vector spaces play an important role. In business, too, the theory of vector spaces is essential. For example, to solve resource allocation problems, we use a method called linear programming, which you will study in Unit 10.

Units 8 and 9 may appear difficult because of their abstraction. If you find this a problem, pick out some specific examples and work with them until you understand the general principles involved. You may also want to reread the discussion of proving theorems in Unit 2, and the examples of proofs in Unit 4.

Objectives

When you have completed this unit, you should be able to

1. define “n-dimensional Euclidean space,” and describe some of the basic properties of n-dimensional Euclidean vectors.
2. state the Cauchy-Schwarz and triangle inequalities for n-dimensional vector spaces from memory.
3. give the definition of orthogonality in n-space, and the n-dimensional version of the Pythogorean theorem.
4. describe the relationship between matrix multiplication and the n-dimensional dot product.
5. give the definition of a linear transformation from Rn to Rm.
6. describe some of the geometric properties of linear transformations in two- and three-dimensional Euclidean space.
7. give the definition of the reflection, projection, rotation, dilation and contraction operators on R2 and R3.
8. form the composition of two different linear transformations.
9. describe some of the general algebraic properties of linear transformations.
10. describe the geometric significance of the eigenvectors of a linear transformation.