Basics

Definitions
Independence
Subspaces, span, affine sets
Basis, dimension

Definitions

Vectors

Assume we are given a collection of real numbers, x_1,ldots,x_n . We can represent them as locations on a line. Alternatively, we can represent the collection as a single point in a -dimensional space. This is the vector representation of the collection of numbers; each number x_i is called a component or element of the vector.

Vectors can be arranged in a column, or a row; we usually write vectors in column format:

$x = left( begin{array}{c} x_1 vdots x_n end{array} right) .$

We denote by $mathbf{R}^n$ denotes the set of real vectors with components. If $x in mathbf{R}^n$ denotes a vector, we use subscripts to denote components, so that x_i is the -th component of . Sometimes the notation x(i) is used to denote the -th component.

A vector can also represent a point in a multi-dimensional space $mathbf{R}^n$ , where each component corresponds to a coordinate of the point.

Example: The vector x = (2,1) in $mathbf{R}^2$ .

Examples:

Transpose

If is a column vector, x^T denotes the corresponding row vector, and vice-versa. Hence, if is the column vector above:

$x^T = left( begin{array}{ccc} x_1 & ldots & x_n end{array} right) .$

Sometimes we use the looser, in-line notation x=(x_1,ldots,x_n) , to denote a row or column vector, the orientation being understood from context.

Matlab syntax

A column vector x=(2,3.1,-4) and its transpose can be declared in Matlab's workspace as follows. Here, no room for loose notation: we use a semicolon to separate the components of a column vector, while we use commas for row vectors.

Matlab syntax: declare and transpose a vector
>> x = [2; 3.1; -4]; % declare a column vector using ";".
>> y = x'; % the prime operator ' transposes the vector.
>> y = [2,3.1,-4]; % can also declare a row vector with commas.
>> x(2) % this produces the second component of x.
>> x([1,3]) % this produces the 2-vector with the first and the third component of x.

Independence

A set of vectors ${x_1,ldots,x_n}$ in ${mathbf{R}}^n$ , i=1,ldots,m is said to be independent if and only if the following condition on a vector $lambda in {mathbf{R}}^m$ :

$sum_{i=1}^m lambda_i x_i = 0$

implies lambda = 0 . This means that no vector in the set can be expressed as a linear combination of the others.

Example: the vectors x^1=[1,2,3] and x^2 = [3,6,9] are not independent, since 3x^1 - x^2 = 0 .

Subspace, span, affine sets

A subspace of ${mathbf{R}}^n$ is a subset that is closed under addition and scalar multiplication. Geometrically, subspaces are ‘‘flat’’ (like a line or plane in 3D) and pass through the origin.

An important result of linear algebra, which we will prove later, says that a subspace mathbf{S} can always be represented as the span of a set of vectors $x_i in {mathbf{R}}^n$ , i=1,ldots,m , that is, as a set of the form

$mathbf{S} = mbox{rm span}(x_1,ldots,x_m) := left{ sum_{i=1}^m lambda_i x_i ~:~ lambda in {mathbf{R}}^m right} .$

An affine set is a translation of a subspace — it is ‘‘flat’’ but does not necessarily pass through , as a subspace would. (Think for example of a line, or a plane, that does not go through the origin.) So an affine set mathbf{A} can always be represented as the translation of the subspace spanned by some vectors:

$mathbf{A} = left{ x_0 + sum_{i=1}^m lambda_i x_i ~:~ lambda in {mathbf{R}}^m right} ,$

for some vectors x_0,x_1,ldots,x_m . In shorthand notation, we write $mathbf{A} = x_0 + mathbf{S}$ .

Example: In $mathbf{R}^3$ , the span mathbf{S} of the two vectors

u = left[ begin{array}{c} -1 2 0.5 end{array} right], ;; v := left[ begin{array}{c} 1 3 0.1 end{array} right]

is the plane passing through the origin pictured in blue.

When mathbf{S} is the span of a single non-zero vector, the set mathbf{A} is called a line passing through the point x_0 . Thus, lines have the form

$left{ x_0 + t u ~:~ t in mathbf{R} right}.$

where determines the direction of the line, and x_0 is a point through which it passes.

Example: A line in $mathbf{R}^2$ passing through the point x_0 = (2,0) , with direction u = (0.8944,0.4472) .

Basis, dimension

Basis in $mathbf{R}^m$

A basis of ${mathbf{R}}^n$ is a set of independent vectors. If the vectors u_1,ldots,u_n form a basis, we can express any vector as a linear combination of the u_i 's:

$x = sum_{i=1}^n lambda_i u_i$

for appropriate numbers lambda_1,ldots,lambda_n .

The standard basis (alternatively, natural basis) in ${mathbf{R}}^n$ consists of the vectors e_i , where e_i 's components are all zero, except the -th, which is equal to . In $mathbf{R}^3$ , we have

$e_1 := left(begin{array}{c} 1 0 0 end{array}right) , ;; e_2 := left(begin{array}{c} 0 1 0 end{array}right) , ;; e_3 := left(begin{array}{c} 0 0 1 end{array}right) .$

Example: A basis in $mathbf{R}^3$ .

Basis of a subspace

The basis of a given subspace $mathbf{S}subseteq{mathbf{R}}^n$ is any independent set of vectors whose span is . If the vectors u_1,ldots,u_r form a basis of , we can express any vector as a linear combination of the u_i 's:

$x = sum_{i=1}^r lambda_i u_i$

for appropriate numbers lambda_1,ldots,lambda_r .

The number of vectors in the basis is actually independent of the choice of the basis (for example, in ${mathbf{R}}^3$ you need two independent vectors to describe a plane containing the origin). This number is called the dimension of mathbf{S} . We can accordingly define the dimension of an affine subspace, as that of the linear subspace of which it is a translation.

Examples:

The dimension of a line is , since a line is of the form $x_0 + mbox{bf span}(x_1)$ for some non-zero vector .
Dimension of an affine subspace.