MA 229 Notes Day 01

Welcome to linear algebra

What is linear algebra? Linear algebra studies a mathematical abstraction called vectors. So what is a vector?

Well, there is a technical definition, but right now that wouldn't be very helpful. Why not? Well, consider the technical definition of something you understand well, namely a circle.

A circle is the set of all points equidistant from a given point.

The above is how mathematics creates a precise, unambiguous, formal definition of a circle. If you wanted to prove that the distance all the way around a circle is always more than 3 times as far as the distance across, you could prove that with the above definition. But critical to understanding circles is understanding what idea they were designed to capture. The circle, of course, is a mathematical abstraction designed to describe "objects that are round".

It's the same way with vectors; rather than put the formal definition of what a vector is up front, it's more helpful to start with this:

A vector is a mathematical abstraction designed to describe objects that can be scaled and added.

Someone trying to understand what a circle is might ask you what you mean by "objects that are round". At that point, you could simply throw up your hands and give the technical definition of a circle. Or you could try to convey the idea of "round" by providing examples of "objects that are round".

In the same way, one might respond to our statement about vectors by asking what we mean by "scale" and "add". Later on, we will have to do some work to be precise about what these things mean. But the right thing to do for now is to give you a sense of these things as ideas, by showing you some examples of "objects that can be scaled and added" — that is, some examples of vectors!

[ Notes: Sometimes you will hear vectors described as "quantities that have both magnitude and direction".]

Examples of Vectors — Abstract examples

Example 1: Arrow vectors

This is a very important example of vectors. You draw an arrow (or a "directed line segment" if you wanted to be all fancy) like so:

[ picture of arrow ]

The pointy end of the arrow is called its "head" (as in "arrowhead") and the other end is called its "tail". (This turns out to be important to remember.) Also, note that where the arrow is drawn makes no difference. For example, the following two arrows are identical:

[ picture or same arrow in two positions]

[Note: You can also have arrows in 3-dimensional space, not just the plane — or even 4- or 5-dimensional space. For now, we follow the guiding principal of sticking to two dimensions for a while.]

Note that the objects themselves are simple! We are already done describing them. However, we still have to describe how they are to be scaled and added.

Scaling

Things we need to make sense of vectors 1: We need a collection of things that we can scale by. They might be integers, or real numbers, or even complex numbers. Each such thing is called a scalar.

To scale a vector $\vec v$ is to combine it with a scalar in a way that produces a new vector. If we have a vector $\vec v$ and a scalar $a$ then we denote the result of scaling $\vec v$ by $a$ as $a \vec v$.

In the case of arrows in the plane, we take our scalars to be real numbers. Given an arrow (vector) and a number (scalar) we get the result of scaling the arrow by that number by simply forming the new arrow whose length is the old length times the new number.

[ example of arrow that gets scaled ]

The only unexpected thing here is what happens when the scalar is negative. We could just disallow scaling by negative numbers. But we choose to pursue another idea instead, namely that scaling by a negative number flips the direction by 180 degrees.

Adding

Given two vectors, to add them is to combine them in some way in order to produce a new vector.

Note there isn't anything about the above that implies our "adding" has to have anything at all to do with "adding" in the sense of numbers, or really any other sense of "adding" you've ever heard before. (In practice, there will always be a relationship, but this isn't something we can see from the get-go.)

The way we add two arrows is not obvious, but seems natural once you see it. Given two arrow vectors $v_1$ and $v_2$, we produce the new vector $v_1 + v_2$ (this is one vector) by drawing $v_1$ anywhere we like, then drawing

  • draw $v_2$ so that its tail coincides with the head of $v_1$
  • draw a new arrow starting at the tail of tail of $v_1$ and ending at the head of $v_2$

[ picture of adding two vectors together ]

Given the geometric way in which we defined the addition of vectors, it's easy to see that the order in which we choose to add two vectors doesn't make any difference. The proof of this is as simple as the following figure:

[ draw both ways of adding the two vectors ]

It turns out that we will always want this to be true.

Things we need to make sense of vectors 2: We need it to be the case that it never matters what order we add two vectors in. That is, whenever we have vectors $u$ and $v$, the vector $u+v$ is the same as the vector $v+u$. The fancy buzzword for this is that addition of vectors is commutative.

Because drawing both ways of adding the two vectors in the picture above results in a picture that looks like a parallelogram, the geometric way in which we add arrow vectors is sometimes called the parallelogram rule.

A good question might be: Is there a way to subtract vectors? There are a couple of ways to answer this question:

  • Answer 1 is that $v_1-v_2$ is the same thing as $v_1 + (-1)v_2$. This is common in mathematics; if we define addition and multiplication by negatives, then "subtraction" falls out for free.
  • Answer 2 is obtained by adding a new vector to our parallelogram picture like so:

[ complete parallelogram picture ]

Notice that the dashed vector that goes across has the property that when added to the vector $v$ it gives the vector $u$. Since we expect this to be true of $u-v$, it follows that if only one vector gets to be $u-v$, it had better be the dashed vector above. We could even geometrically define the vector $u-v$ to be the vector drawn from the head of $v$ to the head of $u$.

Example 2: Coordinate vectors

A coordinate vector is simply a list of numbers considered in a specific order. The $i$th number in the list is called the $i$th coordinate of the vector. If it sounds simple enough, that's because it really is. Before we can say much more however we just need to settle on the way that we would like to actually write such a vector down. Basically, we just list out the numbers one after another either vertically or horizontally and then put a bracket on either side.

Definition: A column vector is a coordinate vector for which the coordinates are written vertically. The 1st coordinate of the vector appears at the top and the final coordinate appears at the bottom. Square brackets or parentheses are written on either side. In these notes, we use square brackets, since those have become more popular lately.

Example of a column vector: $\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}$.

Definition: A row vector is a coordinate vector for which the coordinates are written horizontally, left-to-right. To make things less confusing visually, we write commas in between the coordinates. Square brackets or parentheses are written on either side. In these notes, we use square brackets.

Example of a row vector: $[1,2,3,4]$.

Note: Whenever $u$ is a coordinate vector, we will denote its $i$th coordinate by $u_i$.

Coordinate vectors are very easy to work with, because in order to add two coordinate vectors, one simply adds the corresponding coordinates, while to scale a coordinate vector, one simply multiplies each coordinate by the desired scalar. We present this as a formal definition below.

Formal definition of coordinate vector scaling and adding goes here.

Example of scaling a coordinate vector.

Example of adding two coordinate vectors.

Notice that once again it does not matter what order we add the vectors in!

Examples of vectors — Vectors in applications

In contrast with thinking about vectors that are mathematical abstractions, such as arrow vectors and coordinate vectors, we can also think about "vectors in the real world", or more accurately, real-world phenomena that can be modeled with vectors. As this course goes on, keep an eye out for such things in the world around you and in your other courses. Remember that you are looking for objects that can be both scaled and added. At this early stage, before you have a lot of experience scaling and adding, before we have pinned down exactly what we need to be true of scaling and adding for our theories to work properly, it won't be entirely clear. However, will give you some examples here in this section to get you thinking. Occasionally, as the course goes on, we will return to these examples to check in on how the theory that we are building up can tell us something about them.

Velocity

Displacement

Force

Sound

Color

Taste in movies

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