Basics
Basics
- Definitions
- Independence
- Subspaces, span, affine sets
- Basis, dimension
Definitions
Vectors
Assume we are given a collection of real numbers,
. 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
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:
(1)
We denote by denotes the set of real vectors with
components. If
denotes a vector, we use subscripts to denote components, so that
is the
-th component of
. Sometimes the notation
is used to denote the
-th component.
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A vector can also represent a point in a multi-dimensional space ![]() Example: The vector |
Examples:
Transpose
If is a column vector,
denotes the corresponding row vector, and vice-versa. Hence, if
is the column vector above:

Sometimes we use the looser, in-line notation , to denote a row or column vector, the orientation being understood from context.
Matlab syntax
A column vector 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.
>> 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 in
,
. is said to be independent if and only if the following condition on a vector
:
implies . This means that no vector in the set can be expressed as a linear combination of the others.
Example: the vectors and
are not independent, since
.
Subspace, span, affine sets
A subspace of 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 can always be represented as the span of a set of vectors
,
, that is, as a set of the form

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
can always be represented as the translation of the subspace spanned by some vectors:

for some vectors . In shorthand notation, we write
![]() |
Example: In ![]() ![]() (2) [latex],\begin{align} v &= \begin{bmatrix} 1 \\ 3 \\ 0.1 \end{bmatrix} \end{align}[/latex] is the plane passing through the origin pictured in blue. |
When is the span of a single non-zero vector, the set
is called a line passing through the point
. Thus, lines have the form
where determines the direction of the line, and
is a point through which it passes.
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Example: A line in ![]() ![]() ![]() |
Basis, dimension
Basis in 
A basis of is a set of
independent vectors. If the vectors
form a basis, we can express any vector as a linear combination of the
‘s:

for appropriate numbers .
The standard basis (alternatively, natural basis) in consists of the vectors
, where
‘s components are all zero, except the
-th, which is equal to
. In
, we have
(3)
[latex],\begin{align} e_2 &:= \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \end{align}[/latex]
(4)
Example: A basis in .
Basis of a subspace
The basis of a given subspace is any independent set of vectors whose span is
. If the vectors
form a basis of
, we can express any vector as a linear combination of the
‘s:


The number of vectors in the basis is actually independent of the choice of the basis (for example, in you need two independent vectors to describe a plane containing the origin). This number is called the dimension of
. 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
for some non-zero vector
.
- Dimension of an affine subspace.