What is the definition of a span in linear algebra, and how does it relate to vector spaces?

In linear algebra, the concept of a span is fundamental to understanding the structure of vector spaces. The span of a set of vectors is defined as the collection of all possible linear combinations of those vectors. A linear combination of a set of vectors involves multiplying each vector by a scalar (which can be any real or complex number, depending on the context) and then adding the results together.

To elaborate, let's consider a set of vectors \( \{ \mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_k \} \) in a vector space \( V \). The span of this set, denoted as \( \text{span}(\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_k) \), is the set of all vectors that can be expressed as:

\[
\text{span}(\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_k) = \left\{ c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + ... + c_k\mathbf{v}_k \mid c_1, c_2, ..., c_k \in \mathbb{F} \right\}
\]

where \( \mathbb{F} \) is the field over which the vector space is defined (typically the field of real numbers \( \mathbb{R} \) or complex numbers \( \mathbb{C} \)) and \( c_1, c_2, ..., c_k \) are scalars from that field.

The span of a set of vectors has several important properties:

1. Subspace: The span of any set of vectors in \( V \) is itself a subspace of \( V \). This means that the span is closed under vector addition and scalar multiplication, and it contains the zero vector.

2. Basis and Dimension: If the set of vectors \( \{ \mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_k \} \) is linearly independent and spans the vector space \( V \), then it forms a basis for \( V \). The number of vectors in the basis is called the dimension of the vector space.

3. Linear Independence: A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. If the vectors are linearly dependent, some vectors in the set can be removed without changing the span.

To illustrate the concept of span with an example, let's consider the vector space \( \mathbb{R}^3 \) and two vectors within it, \( \mathbf{v}_1 = (1, 0, 0) \) and \( \mathbf{v}_2 = (0, 1, 0) \). The span of these two vectors is the set of all linear combinations of \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \):

\[
\text{span}(\mathbf{v}_1, \mathbf{v}_2) = \left\{ a(1, 0, 0) + b(0, 1, 0) \mid a, b \in \mathbb{R} \right\} = \left\{ (a, b, 0) \mid a, b \in \mathbb{R} \right\}
\]

This span is a plane through the origin in \( \mathbb{R}^3 \), which contains all points with a zero \( z \)-coordinate. It is a two-dimensional subspace of \( \mathbb{R}^3 \).

Understanding the span of a set of vectors is crucial for solving systems of linear equations, analyzing the structure of vector spaces, and performing operations such as matrix transformations and eigenvalue computations. It is a key concept in fields ranging from pure mathematics to applied disciplines such as physics, engineering, and computer science.