Vectors and the Plane

Everything in this chapter is built out of one object, and it is a very simple one. This note introduces it and the three things you can do with it. If you have met vectors before, skim to the last section, which sets up the notation the rest of the chapter uses.

01.A vector is a list of numbers

A vector is an ordered list of numbers. That is the entire definition.

Write $v = (3, 1)$. This is a vector with two entries, called its components. The number of components is the dimension. So $(3, 1)$ is two-dimensional, $(5, 11, 2)$ is three-dimensional, and a list of $256$ numbers is a $256$-dimensional vector.

The set of all two-dimensional vectors whose entries are real numbers is written $\mathbb{R}^2$. Similarly $\mathbb{R}^3$, and in general $\mathbb{R}^n$. The notation is doing exactly what it looks like: $\mathbb{R}$ from Sets and Notation is the real numbers, and the exponent counts how many of them are in the list.

In two dimensions a vector has a picture. Read $(3, 1)$ as an instruction: go three units right, then one unit up. That lands you at a point, and the vector is drawn as an arrow from the origin to it. The picture is genuinely helpful, and it is also the thing that stops being available in dimension $256$, which is worth remembering every time this chapter draws something.

02.Adding vectors

To add two vectors, add their components separately. Nothing else happens.

$$(3, 1) + (1, 2) \;=\; (3 + 1,\; 1 + 2) \;=\; (4, 3).$$

Geometrically this is the instruction-following you would guess. Walk three right and one up, then from wherever you are walk one right and two up. You end at four right and three up.

Two things follow immediately and will be used constantly. The order does not matter, since $(1,2) + (3,1) = (4,3)$ as well. And the vector $(0, 0)$, called the zero vector, changes nothing when added.

03.Scaling vectors

To multiply a vector by a single number, multiply every component by it. A single number in this context is called a scalar, to distinguish it from a vector.

$$3 \cdot (1, 2) \;=\; (3 \cdot 1,\; 3 \cdot 2) \;=\; (3, 6).$$

Geometrically, scaling by $3$ stretches the arrow to three times its length in the same direction. Scaling by $-1$ flips it to point the opposite way:

$$-1 \cdot (3, 1) \;=\; (-3, -1).$$

Scaling by $0$ collapses any vector to $(0,0)$.

Subtraction is these two operations combined. To compute $v - w$, scale $w$ by $-1$ and add:

$$(3, 1) - (1, 2) \;=\; (3, 1) + (-1, -2) \;=\; (2, -1).$$

04.Linear combinations

Now put both operations together, and you have the only construction this chapter actually needs.

Given two vectors $b_1$ and $b_2$, and two scalars $x_1$ and $x_2$, the vector

$$x_1 b_1 + x_2 b_2$$

is called a linear combination of $b_1$ and $b_2$. Scale each one, add the results.

Take $b_1 = (3, 1)$ and $b_2 = (1, 2)$ and work a few by hand.

With $x_1 = 1$, $x_2 = 0$, the combination is just $b_1 = (3,1)$.

With $x_1 = 1$, $x_2 = 1$:

$$1 \cdot (3,1) + 1 \cdot (1,2) \;=\; (3,1) + (1,2) \;=\; (4,3).$$

With $x_1 = 2$, $x_2 = -1$:

$$2 \cdot (3,1) + (-1) \cdot (1,2) \;=\; (6,2) + (-1,-2) \;=\; (5, 0).$$

With $x_1 = 0.5$, $x_2 = 0.5$:

$$0.5 \cdot (3,1) + 0.5 \cdot (1,2) \;=\; (1.5, 0.5) + (0.5, 1) \;=\; (2, 1.5).$$

Notice that last one. Allowing fractional scalars produced a point with fractional coordinates. Hold on to that observation, because the next note removes it and that removal is the entire subject of the chapter.

05.Bases

If you let $x_1$ and $x_2$ range over all real numbers, the combinations $x_1 b_1 + x_2 b_2$ with $b_1 = (3,1)$ and $b_2 = (1,2)$ produce every single point in the plane. Not most of them. All of them.

That is worth checking rather than believing. Suppose you want to reach $(5, 0)$. The computation above already did it, with $x_1 = 2$ and $x_2 = -1$. Suppose you want $(0, 1)$. Solving

$$3x_1 + x_2 = 0, \qquad x_1 + 2x_2 = 1$$

gives $x_2 = -3x_1$ from the first equation, so $x_1 - 6x_1 = 1$, meaning $x_1 = -1/5$ and $x_2 = 3/5$. Fractional, but perfectly legal.

When a set of vectors can reach every point of the space by linear combination, and no vector in the set is redundant, the set is called a basis. The pair $b_1 = (3,1)$, $b_2 = (1,2)$ is a basis of $\mathbb{R}^2$.

"No vector is redundant" has a precise name, linear independence, and in two dimensions it means exactly one thing: the two vectors do not lie along the same line. If $b_2$ were $(6, 2)$, which is $2 b_1$, then every combination would stay on that one line and the plane would be unreachable.

The most familiar basis of $\mathbb{R}^2$ is $e_1 = (1, 0)$ and $e_2 = (0, 1)$, called the standard basis, because reaching $(a, b)$ needs only $a e_1 + b e_2$ and the coefficients are the coordinates themselves.

Two facts to carry forward. A space has many different bases, not one. And the number of vectors in a basis is always the dimension, so every basis of $\mathbb{R}^2$ has exactly two vectors.

The next note changes one word in the definition of linear combination, and the plane turns into something with gaps in it.