Examples of Vector Spaces of Dimension 6 Other Than ( mathbb{R}^6 ) or ( mathbb{C}^6 )

A vector space of dimension 6 can be defined as a space with a basis consisting of 6 linearly independent vectors. While the most straightforward example might be ( mathbb{R}^6 ) or ( mathbb{C}^6 ), there are numerous other vector spaces with the same dimension, each with unique properties and structures. Below are some illustrative examples of such vector spaces.

1. Vector Space of Polynomials of Degree at Most 5

Consider the vector space ( P_5 ) of all polynomials with real coefficients of degree at most 5. This vector space can be represented as:

P_5  {a   bx   cx^2   dx^3   ex^4   fx^5 : a, b, c, d, e, f in mathbb{R}}

This space has a basis consisting of the set of polynomials ( {1, x, x^2, x^3, x^4, x^5} ). Hence, ( P_5 ) is a vector space of dimension 6, perfectly illustrating that a vector space can have a different structure even though it has the same dimension as ( mathbb{R}^6 ).

2. Vector Space of 6x1 Column Matrices with a Linear Condition

Consider the vector space of all 6x1 column matrices (column vectors) with real entries, denoted by ( mathbb{R}^6 ). If we impose a linear condition on these matrices such as ( x_1 x_2 x_3 x_4 x_5 x_6 0 ), the resulting subspace is a 5-dimensional vector space. This subspace is a non-standard example of a 6-dimensional space but with a specific structure imposed by the condition.

3. Direct Sum of Two 3-dimensional Vector Spaces

The direct sum of two vector spaces, such as ( mathbb{R}^3 oplus mathbb{R}^3 ), is another example. This space has a dimension of 6 but is specifically structured as the direct sum of two 3-dimensional spaces. Mathematically, it can be represented as:

mathbb{R}^3 oplus mathbb{R}^3  {(a, b, c) oplus (d, e, f) : a, b, c, d, e, f in mathbb{R}}

Each element in this space can be viewed as a pair of 3-dimensional vectors, making the overall structure distinct from ( mathbb{R}^6 ).

4. Vector Space of Symmetric and Skew-Symmetric Matrices

For more advanced examples, we turn to the vector spaces of symmetric and skew-symmetric matrices. Symmetric Case: For a symmetric matrix of size ( n times n ) over the field ( mathbb{F} ), the set of all such matrices forms a vector space. The dimension of this space is given by ( frac{n(n 1)}{2} ). When ( n 3 ) and ( mathbb{F} mathbb{R} ), the vector space of symmetric 3x3 matrices is of dimension 6. Skew-Symmetric Case: Similarly, for a skew-symmetric matrix of size ( n times n ) over the field ( mathbb{F} ), the set of all such matrices also forms a vector space, and its dimension is ( frac{n(n-1)}{2} ). When ( n 4 ) and ( mathbb{F} mathbb{R} ), the vector space of skew-symmetric 4x4 matrices is of dimension 6.

Conclusion

These examples illustrate that there are many vector spaces of dimension 6, each with its unique structural and algebraic properties. While the most common examples might be ( mathbb{R}^6 ) or ( mathbb{C}^6 ), the space of polynomials, the space of column matrices with a linear condition, the direct sum of two 3-dimensional spaces, and vector spaces of symmetric and skew-symmetric matrices are just a few of the myriad ways to construct such spaces. Understanding these examples can provide valuable insights into the broader theory of vector spaces and linear algebra.