The Dimension of a Vector Space Basis

The dimension of a vector space basis is the number of linearly independent vectors {v₁,...,v_n} in the basis.
$$ dim_k(n) $$ $$with \:\: n \in Z >=0 $$

The dimension n (or cardinality) is a non-negative integer ranging from zero to infinity.

It is denoted as dim_k(n), where K is the field of the vector space, and n represents the cardinality.

Note. If the field K is implicit, it can be omitted, and the dimension simply indicated as dim(n).

The dimension of a vector space basis can be either finite or infinite.

• Finite Dimension
  The basis has a finite dimension equal to a finite number n, meaning it consists of n vectors.
  $$ B = \{ v_1 , ... , v_n \} $$
• Infinite Dimension
  The basis has an infinite dimension, meaning it consists of an infinite number of vectors.
  $$ B = \{ v_1 , v_2 , ... \} $$

Practical Examples

Example 1

The following vector space basis consists of two elements, namely two vectors v₁ and v₂, therefore it has a dimension of 2.

$$ B = \{ v_1 , v_2 \} $$

Example 2

This vector space basis is composed of three vectors {v₁,v₂,v₃}.

Hence, the basis has a dimension of 3.

$$ B = \{ v_1 , v_2, v_3 \} $$

Example 3

This set of vectors is a trivial space {0_v}.

It consists only of the zero vector and is not a basis. Therefore, it has a dimension of 0.

$$ B = \{ 0_v \} $$

Zero Dimension and the Trivial Space

The zero dimension (or null) is a special case of finite dimension.

In vector space, only the trivial space has a zero dimension because it is solely comprised of the zero vector 0_v.

$$ \{ 0_v \} $$

The zero vector 0_v is linearly dependent.

Note. A vector is linearly independent if a zero vector can only be obtained by setting all coefficients k₁,k₂,...,k_n=0 of the linear combination to zero. $$ \vec{v} = k_1 \cdot \vec{v}_1 + k_2 \cdot \vec{v}_2 + ... + k_n \cdot \vec{v}_n $$ However, in the case of the zero vector, setting any of the coefficients to a non-zero value k_n≠0 still results in the zero vector. Therefore, the zero vector does not meet the definition of a linearly independent vector. It is always linearly dependent.

As a result, the trivial space cannot be a vector space basis.

The trivial space is the only vector space with a zero dimension.

And so on