Linear Vector Spaces: Basic Definitions
Linear Vector Space
A set is called a linear vector space over the field of if the two operations (vector addition) and (scalar multiplication) are defined and satisfy the following 8 axioms: :
- Distributivity of vector addition over scalar multiplication:
- Distributivity of scalar addition over scalar multiplication:
- Compatibility of scalar multiplication with field multiplication:
- Identity element of addition/Zero element: such that
- Inverse element of addition: such that . is denoted
- Identity element of scalar multiplication:
The elements of the linear vector space are called vectors, while the elements of are called scalars. In general, is either the field of real numbers or of complex numbers . For the remainder of this article, is assumed to be the field of real numbers unless otherwise specified.
In the history of the development of linear vector spaces, the following statements sometimes appeared in the definition. However, the above 8 essential axioms can be used to prove them.
- The element is unique.
- is unique.
- If , then .
- and .
Notice that in the above definition and statements, the hat symbol was used to distinguish the zero vector from the zero scalar . However, in the future, will be used for both and it is to be understood from the context whether it is the zero vector or the zero scalar.
Examples of linear vector spaces include the following sets:
- The set is a linear vector space!
A subset of a vector space is called a subspace if .
In other words, a subset of a vector space is called a subspace if it is closed under both vector addition and scalar multiplication. For example, the set is a subspace of (why?).
Let be a linear vector space over . A set of vectors is called a linearly independent set of vectors if none of the vectors can be expressed linearly in terms of the other elements in the set; i.e., the only linear combination that would produce the zero vector is the trivial combination:
Otherwise, the set is called linearly dependent.
Consider the vector space . The set where and is linearly independent:
However, the set where and is linearly dependent:
This means that there are multiple combinations that would give the zero vector. For example, choosing and would yield .
Notice that if the zero vector is an element of a set then, the set is automatically linearly dependent since the following non trivial combination would produce the zero vector:
Basis and Dimensions
A subset of a linear vector space is called a basis set if it is a set of linearly independent vectors such that can be expressed as a linear combination of the elements of . Alternatively: is a basis of if and only if the following two conditions are satisfied:
- such that
The dimension of a linear vector space is the number of elements in its basis set.
For example is a two dimensional space, because its basis set has two elements and .
Assertion: If is a basis set of , then the combination is unique. i.e., such that .
Proof: Use the linear independence of the basis set to prove the result.
Click on the image below to interact with this example. Input the components of the vectors and . The tool draws the vectors and in blue and red respectively. It identifies if the vectors are linearly dependent or linearly independent. If they are linearly independent, the tool calculates and draws the components of in the directions of and .