Subspaces, span, affine sets
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 of the vector.
Vectors can be arranged in a column, or a row; we usually write vectors in column format: 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 we use the in-line notation .
Example: The vector in .
If is a column vector, denotes the corresponding row vector, and vice-versa. Hence, if is the column vector above:
A column vector and its transpose can be declared in Matlab’s workspace as follows. We can also declare as a row vector directly.Matlab syntax
>> x = [2; 3.1; -4]; % declare a column vector using ; >> y = x'; % ' transposes the vector >> y = [2,3.1,-4]; % can also declare a row vector with commas.
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 .
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. A subspace can always be represented as the span of a set of vectors , , which is the set
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 , the span of the two vectors 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.
Example: A line in passing through the point , with direction .
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
Example:A basis in .
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: for appropriate numbers .
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 it is a translation of.
The dimension of a line is , since a line is of the form for some non-zero vector .