What is a Basis in Linear Algebra?
What is the basis?
In linear algebra, a basis is a set of vectors that span a space. It is one of the most important concepts in generating vector spaces and linear transformations. It is also the base for a number of other concepts such as dimension and rank, which are essential for understanding eigen decomposition and SVD.
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The number of vectors in a basis is called the dimension of the space because each vector has a dimension (the number of coordinates in the space). Many vector spaces have a very direct concept of dimension: Rn has a very clear idea of how many coordinates there are in it. However, it is not always that easy to understand the concept of dimension in other spaces.
For example, a basis for c 00 display style c_00 is made up of the sequences x = an (x n) displaystyle x n / c n displaystyle x n+c n / sup n of real numbers that have only finitely many non-zero elements. It is therefore a countable Hamel basis.
A base can contain only a finite number of vectors, and each of the vectors in the basis must be independent of all other vectors in the space. It is this ability to be linearly independent that allows us to define change of basis and eigendecomposition.
It is important to note that all bases for a given space must have the same number of vectors, and no nontrivial subspace has any other basis. This was first proved by Georg Hamel and was subsequently reaffirmed as the basis extension theorem (the Steinitz exchange lemma).
As you will learn in Chapters 9 and 10 of Essential Math for Data Science, a change of basis is a way to transform vectors from one basis to another. This is fundamental to the concept of eigendecomposition and SVD because it allows you to manipulate your basis matrix so that the effects of any underlying linear transformation are translated into a new basis without changing the basis itself.
Moreover, it is necessary to remember that a change of basis must be invertible and the inverse of the original basis must be a basis of the new space. This is crucial to proving that an injective linear transformation is a bijection and can be used to calculate the matrix of an underlying basis.
In addition to its role in the connection between linear transformations and matrices, a basis also provides important properties of the space under a basis. This property is often referred to as the ”number of basis elements”, and it is an important concept for understanding a variety of basic mathematical properties such as the dimension and rank of a vector space.
The number of components in a vector relative to a basis is the set of uniquely determined scalar coefficients in the component of v with respect to the basis B. This is denoted v B and sometimes written as [v] B.