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Chapter1Vectors: Algebra and Geometry

Primary Goals

The language of vectors is convenient for doing linear algebra. In this chapter, you should learn:

  1. What vectors are, and how to do algebra and geometry with vectors;
  2. What it means to be a linear combination of given vectors;
  3. The concept of the span of some vectors, i.e., the set of all linear combinations of these vectors.

Linear algebra is the study of one or more linear equations, that is, equations that look like

2 x + 3 y + 5 z = 1.

The basic case of one linear equation in one variable, such as 2 x = 5, is not very interesting. We know that there is a unique solution, and we know how to find it: x = 2.5. The complications in linear algebra arise because we consider several simultaneous equations in several variables, for instance

ADCDB 3 x 1 + 4 x 2 + 10 x 3 + 19 x 4 − 2 x 5 − 3 x 6 = 1417 x 1 + 2 x 2 − 13 x 3 − 7 x 4 + 21 x 5 + 8 x 6 = 2567 − x 1 + 9 x 2 + 3 2 x 3 + x 4 + 14 x 5 + 27 x 6 = 26 1 2 x 1 + 4 x 2 + 10 x 3 + 11 x 4 + 2 x 5 + x 6 = − 15.

In order to discuss systems of equations like this, it will be convenient for us to have notation for keeping track of multiple numbers at once. Vectors provide us with this notation. In the case of vectors with 2 or 3 entries, there is a strong connection between vector algebra and geometry. This can be very helpful to us as we develop an intuition for vector algebra, even when there are more than 3 entries. We introduce vectors and discuss the associated geometry in Section 1.1.

In Section 1.2 we introduce the concept of the span of a set of vectors. This is geometric concept, at least for vectors with 2 or 3 entries, that sets the groundwork for answering the most fundamental question about a system of linear equations: does it have any solutions?