Objectives
- Learn how to add and scale vectors in both algebraically and geometrically.
- Understand linear combinations geometrically.
- Pictures: vector addition, vector subtraction, linear combinations.
- Vocabulary: vector, linear combination.
We use to denote the set of all real numbers, i.e., the number line. This contains numbers like
Let be a positive whole number. We define
An -tuple of real numbers is called a point of
In other words, is just the set of all (ordered) lists of real numbers. We will draw pictures of in a moment, but keep in mind that this is the definition. For example, and are points of
When we just get back: Geometrically, this is the number line.
In any the point corresponding to the -tuple is special. We call it the origin.
The elements of are lists of numbers. When is 2 or 3, we can represent these lists geometrically in two ways.
When we can think of as the -plane. We can do so because every point on the plane can be represented by an ordered pair of real numbers, namely, its - and -coordinates.
When we can think of as the space we (appear to) live in. We can do so because every point in space can be represented by an ordered triple of real numbers, namely, its -, -, and -coordinates.
A vector is an element of especially when drawn as an arrow.
The difference is purely psychological: points and vectors are both just representations of lists of numbers.
Geometrically (in 2- or 3-dimensional space) a vector is determined by its direction and its length.
Many physical quantities have a direction and magnitude. Common examples include velocity, momentum, acceleration and force. The language of vectors is good for describing such quantities.
Another way to think about a vector is as a difference between two points, or the arrow from one point to another. For instance, is the arrow from to
When a vector is considered as an arrow from the origin to a point we say is based. In this case, the coordinates of and are the same.
Even though we can't draw elements of geometrically, we will still refer to them as points or vectors. We have to rely more on algebra than on geometric intuition when we work with vectors with more than coordinates.
When we think of an element in as a vector, we will usually write it vertically:
This is called column vector notation. You can also call the notation row vector notation, even though it is the same as ordinary point notation.
We will write for the vector of all s, corresponding to the origin.
So what is or or These are harder to visualize, so you have to go back to the definition: is the set of all ordered -tuples of real numbers
They are still “geometric” spaces, in the sense that our intuition for and often extends to
We will make definitions and state theorems that apply to any but we will only draw pictures for and
The power of using these spaces is the ability to label various objects of interest, such as geometric objects and solutions of systems of equations, by the points of
In the above examples, it was useful from a psychological perspective to replace a list of four numbers (representing traffic flow) or of 841 numbers (representing a QR code) by a single piece of data: a point in some This is a powerful concept; starting in Section 1.2, we will almost exclusively record solutions of systems of linear equations in this way.
Here we learn how to add vectors together and how to multiply vectors by numbers, both algebraically and geometrically.
Addition and scalar multiplication work in the same way for vectors in
Geometrically, the sum of two vectors is obtained as follows: place the tail of at the head of Then is the vector whose tail is the tail of and whose head is the head of Doing this both ways creates a parallelogram. For example,
Why? The width of is the sum of the widths, and likewise with the heights.
Geometrically, the difference of two vectors is obtained as follows: place the tail of and at the same point. Then is the vector from the head of to the head of For example,
Why? If you add to you get
A scalar multiple of a vector has the same (or opposite) direction, but a different length. For instance, is the vector in the direction of but twice as long, and is the vector in the opposite direction of but half as long. Note that the set of all scalar multiples of a (nonzero) vector is a line.
We can add and scale vectors in the same equation.
Let be scalars, and let be vectors in The vector in
is called a linear combination of the vectors with weights or coefficients
Geometrically, a linear combination is obtained by stretching / shrinking the vectors according to the coefficients, then adding them together using the parallelogram law.
A linear combination of a single vector is just a scalar multiple of So some examples include
The set of all linear combinations is the line through . (Unless in which case any scalar multiple of is again )
The set of all linear combinations of the vectors
is the line containing both vectors.
The difference between this and a previous example is that both vectors lie on the same line. Hence any scalar multiples of lie on that line, as does their sum.