Basis
Definition and Overview
In the realm of linear algebra, the concept of a "basis" is fundamental to understanding vector spaces. A basis of a vector space is a set of vectors that are linearly independent and span the entire vector space. This means that any vector in the space can be expressed as a linear combination of the basis vectors. The number of vectors in the basis is called the dimension of the vector space.
The importance of a basis lies in its ability to provide a coordinate system for the vector space, allowing for the representation of vectors in terms of these coordinates. This is crucial in various applications, including solving systems of linear equations, performing transformations, and analyzing vector spaces in higher dimensions.
Properties of a Basis
Linearly Independent
A set of vectors is said to be linearly independent if no vector in the set can be written as a linear combination of the others. This property ensures that each vector in the basis contributes uniquely to the span of the vector space. If a set of vectors is not linearly independent, it cannot serve as a basis.
Spanning the Vector Space
For a set of vectors to be a basis, it must span the vector space, meaning that any vector in the space can be expressed as a linear combination of the basis vectors. This property guarantees that the basis provides a complete description of the vector space.
Uniqueness of Representation
One of the key features of a basis is that it allows for a unique representation of every vector in the vector space. Given a basis \(\{v_1, v_2, \ldots, v_n\}\), any vector \(v\) in the space can be uniquely expressed as:
\[ v = a_1v_1 + a_2v_2 + \cdots + a_nv_n \]
where \(a_1, a_2, \ldots, a_n\) are scalars.
Types of Bases
Standard Basis
In \(\mathbb{R}^n\), the standard basis is the set of vectors \(\{e_1, e_2, \ldots, e_n\}\), where each \(e_i\) is a vector with a 1 in the \(i\)-th position and 0s elsewhere. This basis is particularly useful because it simplifies many calculations and provides an intuitive understanding of the vector space.
Orthonormal Basis
An orthonormal basis is a basis where all vectors are orthogonal to each other and have unit length. This type of basis is advantageous in simplifying computations, especially in the context of inner product spaces. The Gram-Schmidt process is a common method for converting any basis into an orthonormal basis.
Eigenbasis
An eigenbasis is a basis consisting of eigenvectors of a linear transformation. This basis is particularly useful in diagonalization of matrices, where the matrix can be represented in a simpler form, making it easier to compute powers and exponentials of the matrix.
Applications of Bases
Coordinate Systems
Bases are essential in defining coordinate systems. In any vector space, once a basis is chosen, every vector can be represented as a coordinate vector relative to this basis. This representation is crucial in various fields, including physics, engineering, and computer graphics.
Solving Linear Systems
In the context of solving systems of linear equations, a basis provides a framework for expressing solutions. For instance, in homogeneous systems, the solution set can be described as a linear combination of basis vectors of the null space.
Transformations and Projections
Bases play a significant role in linear transformations and projections. When a linear transformation is applied to a vector space, the effect on the basis vectors determines the transformation's matrix representation. Similarly, projections onto subspaces can be easily computed using an orthonormal basis.
Changing Bases
Changing from one basis to another is a common operation in linear algebra. The process involves finding the transition matrix, which relates the coordinates of vectors in the old basis to those in the new basis. This is particularly useful in applications where different bases are more suitable for different tasks.
Transition Matrix
The transition matrix from one basis \(\{v_1, v_2, \ldots, v_n\}\) to another basis \(\{w_1, w_2, \ldots, w_n\}\) is an invertible matrix that transforms coordinates in the old basis to coordinates in the new basis. It is constructed by expressing each vector of the new basis as a linear combination of the old basis vectors.
Example of Basis Change
Consider the vector space \(\mathbb{R}^2\) with the standard basis \(\{e_1, e_2\}\) and another basis \(\{v_1, v_2\}\). If \(v_1 = 2e_1 + e_2\) and \(v_2 = e_1 - e_2\), the transition matrix from the standard basis to \(\{v_1, v_2\}\) is:
\[ \begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix} \]
Basis in Infinite Dimensional Spaces
In infinite-dimensional vector spaces, such as function spaces, the concept of a basis extends to include infinite sets of vectors. Such bases are crucial in functional analysis and the study of Hilbert spaces.
Schauder Basis
A Schauder basis is a type of basis used in infinite-dimensional spaces where every vector can be expressed as a convergent series of basis vectors. This is a generalization of the finite-dimensional basis concept and is essential in the study of Banach spaces.
Hamel Basis
A Hamel basis is a set of vectors in an infinite-dimensional space that is linearly independent and spans the space, similar to the finite-dimensional case. However, unlike a Schauder basis, a Hamel basis does not require convergence, making it less practical for analysis.