PLANE ALGEBRA

1. PLANE ALGEBRA No, it’s not a spelling error, I mean plane, not plain. Plain algebra is: or . We are going to do “algebra of the plane.” We begin, as usual, by fixing some terminology. The plane we will call because, thanks to René Descartes, we can think of it as ordered pairs of real numbers

2. (what does ordered mean?). That’s right, it means that order matters, so the pair is distinct from the pair . For later purposes of useful arrangement, we write the two real numbers vertically (as a 2x1 matrix) and call it a vector. Please do NOT read any additional meaning into the word vector, other than avertical arrangement (two rows, one column) of two real numbers.

3. Pictorial and physical interpretations of these 2x1 matrices called vectors will come later, we are playing with algebra, not geometry (for now!) What can we reasonably do with these pairs of numbers? Here are the usual symbols for algebraic operations: With the knowledge acquired through the experience of past mathematicians, we know that the and can be easily extended, the can also be extended, with limitations, and for now we’ll forget about . The appropriate definitions are:

4. Definition. (The and the ). Let and be any two vectors. We define to be the vector . (The ) For any real number and any vector we define to be the vector

5. Some remarks are important here: • Sums and differences of vectors are vectors. • A real number times a vector is a vector. In this context (remember our discussion?) the real number we multiply by is called ascalar. • There is nothing to limit ourselves to vectors consisting of two real numbers, we most certainly could use three real numbers (we will call that collection of vectors ) or four real numbers ( ) or even numbers ( ! ) (See p. 27 of your textbook)

6. From now on, at least in our undetermined discussions (the undetermined is how many numbers comprise a vector, the we men-tioned before), we will use lower-case, end-of-the-alphabet symbols like to represent vectors. One of the most important concepts about vectors is that of a . The definition is important enough that we give it in the next slide.

7. Definition. Let be k vectors. A linear combination of is any vector that can be written as (in short-hand sigma notation …. ready? ) for some k scalars . Let’s do some examples. We’ll take three vectors in and write a couple of combinations.

8. OK, the examples we just did show that it’s child’s play to construct vectors that are linear combi-nations of a given set of vectors. BUT … Given k vectors and given another vector , can we decide whether or not the given vector is or is not a linear combi-nation of the k vectors ? Let’s work with for a while. Here are three vectors in (also called 3-dimensional vectors)

9. and here is another three-dimensional vector Question: can be written as a linear combination of ? Let’s see … we need to see if we can find three numbers (scalars) so that Applying our definitions of scalar multiplication and vector addition we get the “equation”

10. (OK, I’ll drop the silliness, who becomes x .) that we would like to solve for . Whoa, this is just the linear system I have a nice package I can use to solve it! (There should be infi-nitely many solutions!)

11. Time to look at some pictures. Thanks to René Descartes we know that two-dimensional vectors are just points in the plane, in . (More to come later)