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  1. Learn how to add and scale vectors in R n , both algebraically and geometrically.
  2. Understand linear combinations geometrically.
  3. Pictures: vector addition, vector subtraction, linear combinations.
  4. Vocabulary: vector, linear combination.

Subsection1.1.1Vectors in R n

We use R to denote the set of all real numbers, i.e., the number line. This contains numbers like 0, 3 2 , π ,104,...


Let n be a positive whole number. We define

R n = allordered n -tuplesofrealnumbers ( x 1 , x 2 , x 3 ,..., x n ) .

An n -tuple of real numbers is called a point of R n .

In other words, R n is just the set of all (ordered) lists of n real numbers. We will draw pictures of R n in a moment, but keep in mind that this is the definition. For example, ( 0, 3 2 , π ) and ( 1, 2,3 ) are points of R 3 .

Example(The number line)

When n = 1, we just get R back: R 1 = R . Geometrically, this is the number line.

3 2 1 0 1 2 3

In any R n , the point corresponding to the n -tuple ( 0,0,...,0 ) is special. We call it the origin.

Subsection1.1.2Vectors and geometry

The elements of R n are lists of n numbers. When n is 2 or 3, we can represent these lists geometrically in two ways.

Example(The Euclidean plane)

When n = 2, we can think of R 2 as the xy -plane. We can do so because every point on the plane can be represented by an ordered pair of real numbers, namely, its x - and y -coordinates.

( 1,2 ) ( 0, 3 )

When n = 3, we can think of R 3 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 x -, y -, and z -coordinates.

( 1, 1,3 ) ( 2,2,2 )
Vectors and Points

A vector is an element of R n , especially when drawn as an arrow.

thevector A 13 B

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, A 12 B is the arrow from ( 1,1 ) to ( 2,3 ) .

( 1,1 ) ( 2,3 ) A 12 B
Based vectors

When a vector v is considered as an arrow from the origin to a point p , we say v is based. In this case, the coordinates of v and p are the same.

Even though we can't draw elements of R 4 , R 5 ,... 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 3 coordinates.

When we think of an element in R n as a vector, we will usually write it vertically:

v = E 13 F .

This is called column vector notation. You can also call the notation ( 1,3 ) row vector notation, even though it is the same as ordinary point notation.

We will write 0 for the vector of all 0 s, corresponding to the origin.

So what is R 4 ? or R 5 ? or R n ? These are harder to visualize, so you have to go back to the definition: R n is the set of all ordered n -tuples of real numbers ( x 1 , x 2 , x 3 ,..., x n ) .

They are still “geometric” spaces, in the sense that our intuition for R 2 and R 3 often extends to R n .

We will make definitions and state theorems that apply to any R n , but we will only draw pictures for R 2 and R 3 .

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 R n .

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 R n . This is a powerful concept; starting in Section 1.2, we will almost exclusively record solutions of systems of linear equations in this way.

Subsection1.1.3Vector Algebra and Geometry

Here we learn how to add vectors together and how to multiply vectors by numbers, both algebraically and geometrically.

Vector addition and scalar multiplication

  • We can add two vectors together provided they have the same number of entries:
    C abc D + C xyz D = C a + xb + yc + z D .
  • We can multiply, or scale, a vector by a real number c :
    c C xyz D = C c · xc · yc · z D .
    We call c a scalar to distinguish it from a vector. If v is a vector and c is a scalar, then cv is called a scalar multiple of v .

Addition and scalar multiplication work in the same way for vectors in R n .

The Parallelogram Law for Vector Addition

Geometrically, the sum of two vectors v , w is obtained as follows: place the tail of w at the head of v . Then v + w is the vector whose tail is the tail of v and whose head is the head of w . Doing this both ways creates a parallelogram. For example,

E 13 F + E 42 F = E 55 F .

Why? The width of v + w is the sum of the widths, and likewise with the heights.

v w w v v + w 5 = 1 + 4 = 4 + 1 5 = 2 + 3 = 3 + 2
Vector Subtraction

Geometrically, the difference of two vectors v , w is obtained as follows: place the tail of v and w at the same point. Then v w is the vector from the head of w to the head of v . For example,

E 14 F E 42 F = E 32 F .

Why? If you add v w to w , you get v .

v w v w
Scalar Multiplication

A scalar multiple of a vector v has the same (or opposite) direction, but a different length. For instance, 2 v is the vector in the direction of v but twice as long, and 1 2 v is the vector in the opposite direction of v , but half as long. Note that the set of all scalar multiples of a (nonzero) vector v is a line.

Somemultiplesof v . v 2 v 1 2 v 0 v Allmultiplesof v .

Subsection1.1.4Linear Combinations

We can add and scale vectors in the same equation.


Let c 1 , c 2 ,..., c k be scalars, and let v 1 , v 2 ,..., v k be vectors in R n . The vector in R n

c 1 v 1 + c 2 v 2 + ··· + c k v k

is called a linear combination of the vectors v 1 , v 2 ,..., v k , with weights or coefficients c 1 , c 2 ,..., c k .

Geometrically, a linear combination is obtained by stretching / shrinking the vectors v 1 , v 2 ,..., v k according to the coefficients, then adding them together using the parallelogram law.

Figure29Linear combinations of two vectors in R 2 : move the sliders to change the coefficients of v 1 and v 2 . Note that any vector on the plane can be obtained as a linear combination of v 1 , v 2 with suitable coefficients.
Example(Linear Combinations of a Single Vector)

A linear combination of a single vector v = A 12 B is just a scalar multiple of v . So some examples include

v = E 12 F ,3 2 v = E 3 / 23 F , 1 2 v = E 1 / 2 1 F ,...

The set of all linear combinations is the line through v . (Unless v = 0, in which case any scalar multiple of v is again 0. )

Example(Linear Combinations of Collinear Vectors)

The set of all linear combinations of the vectors

v 1 = E 22 F and v 2 = E 1 1 F

is the line containing both vectors.

v 1 v 2

The difference between this and a previous example is that both vectors lie on the same line. Hence any scalar multiples of v 1 , v 2 lie on that line, as does their sum.