Basis, Dimension, and Coordinate Representation

Introduction

In linear algebra, the concepts of basis, dimension, and coordinate representation are foundational to understanding vector spaces and their transformations. A basis is a set of vectors that spans a vector space, meaning any vector in the space can be expressed as a linear combination of the basis vectors. The dimension of a vector space is the number of vectors in its basis, providing a measure of its "size." Coordinate representation involves expressing a vector in terms of a given basis, which is critical for solving systems of equations, analyzing transformations, and understanding geometric interpretations. These concepts are interrelated and form the backbone of linear algebra, enabling the analysis of abstract vector spaces through concrete examples.

Core Content

1. Definition and Properties of Basis

A basis of a vector space is a maximal set of linearly independent vectors that span the space. Formally, a set of vectors ${ \mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n }$ is a basis if:

  1. Spanning: Every vector in the space can be expressed as a linear combination of the basis vectors.
  2. Linear Independence: No vector in the set can be expressed as a linear combination of the others.
  3. Maximality: The set is maximal in the sense that adding any additional vector would violate linear independence.

For example, in $\mathbb{R}^3$, the standard basis consists of vectors ${ \mathbf{e}_1 = \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix}, \mathbf{e}_2 = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}, \mathbf{e}_3 = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} }$ spans the space and is linearly independent. In an infinite-dimensional space, such as the space of all polynomials of degree ≤ 2, a basis might consist of ${ 1, x, x^2 }$, which spans the space and is linearly independent.

The properties of a basis ensure that the vector space is uniquely determined by its basis, allowing for the construction of coordinate systems. A basis's ability to span the space guarantees that any vector can be represented, while its linear independence ensures that each vector can be uniquely expressed in terms of the basis.

2. Understanding Dimension

The dimension of a vector space is the number of vectors in its basis. This concept is central to linear algebra, as it quantifies the "complexity" of the space. For finite-dimensional spaces, the dimension is equal to the number of vectors in any basis, and it is also the maximum number of linearly independent vectors in the space.

In finite-dimensional vector spaces, the dimension is often denoted as $\dim(V)$, where $V$ is the space. For example, $\dim(\mathbb{R}^n) = n$, as the standard basis has $n$ vectors. However, in infinite-dimensional spaces, such as the space of continuous functions on $[a, b]$, the dimension is not necessarily finite. For instance, the space of all polynomials of degree ≤ 2 has dimension 3, even though the number of coefficients is infinite.

The dimension also plays a crucial role in determining the number of solutions to linear systems. In an $n \times n$ system, if the rank of the coefficient matrix is $n$, the system has a unique solution, and the dimension of the solution space is $n - \text{rank}(A)$. This relationship underscores the importance of dimension in understanding the structure of vector spaces.

3. Coordinate Representation

Coordinate representation involves expressing a vector in terms of a given basis. Given a basis ${ \mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n }$, any vector $\mathbf{u}$ in the space can be written as $\mathbf{u} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_n \mathbf{v}_n$, where $c_i$ are scalars. These scalars $c_1, c_2, \dots, c_n$ are called the coordinates of $\mathbf{u}$ with respect to the basis.

The coordinate representation is unique due to the linear independence of the basis vectors. This property ensures that each vector in the space has a unique coordinate expression in terms of the basis. For example, in $\mathbb{R