Review of Vector Spaces

The basic algebraic structure of linear algebra; a module over a field, such that its elements can be added together or scaled by elements of the field.

In this unit, we will revisit the fundamental concept of vector spaces, which forms the backbone of many areas in mathematics, including topology. We will cover the basic definitions, properties, and examples of vector spaces, and delve into the concepts of subspaces, linear combinations, spans, linear independence, basis, dimension, and inner product spaces.

Definition and Examples of Vector Spaces

A vector space (also called a linear space) is a collection of objects called vectors, which can be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.

Examples of vector spaces include sets of vectors in n-dimensional Euclidean space, all polynomials of a certain degree (or less), all m x n matrices, and more.

Subspaces, Linear Combinations, Spans

A subspace of a vector space is a subset of the vector space that is itself a vector space, under the operations of vector addition and scalar multiplication as they are defined on the larger space.

A linear combination of some vectors is an expression obtained from these vectors by multiplying each vector by a scalar and adding the results. The span of a set of vectors is the set of all of its linear combinations.

Linear Independence and Dependence

A set of vectors is said to be linearly independent if no vector in the set can be defined as a linear combination of the others. If a set of vectors is not linearly independent, then it is linearly dependent.

Basis and Dimension of a Vector Space

A basis of a vector space is a set of linearly independent vectors that span the full vector space. Every vector in the space can be expressed uniquely as a linear combination of the basis vectors.

The dimension of a vector space is the number of vectors in any basis for the space. It is a measure of the "size" of the space.

Inner Product Spaces, Orthogonality

An inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of vectors and the angle between vectors.

Orthogonality is a generalization of the concept of perpendicular vectors. Two vectors are orthogonal if their inner product is zero.

In the next unit, we will build on these concepts to explore linear operators and functionals.