Span in Linear Algebra

A span of a set of vectors is the set of all their linear combinations. It represents all vectors that can be formed by scaling and adding the given vectors.

For example, two non-collinear vectors in 2D span the entire plane, while collinear vectors span only a line. In 3D, three non-coplanar vectors span the entire space, and coplanar vectors span a plane.

If the vectors are \{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \}, then their span is

c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \cdots + c_n \mathbf{v}_n

where c_1, c_2, \ldots, c_n are scalars.

Some examples to illustrate the concept of span:

Example 1: Span of Two Vectors in \mathbb{R}^2

Consider two vectors \mathbf{v}_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \text{ and } \mathbf{v}_2 = \begin{pmatrix} 3 \\ 4 \end{pmatrix}

The span of \{\mathbf{v}_1, \mathbf{v}_2\} is the set of all vectors that can be written as:

c_1\mathbf{v}_1 + c_2\mathbf{v}_2 = c_1\begin{pmatrix} 1 \\ 2 \end{pmatrix} + c_2 \begin{pmatrix} 3 \\ 4 \end{pmatrix}

This can be written as:

\begin{aligned}\text{Span}\{\mathbf{v}_1, \mathbf{v}_2\}&= \left\{\begin{pmatrix}c_1 + 3c_2 \\2c_1 + 4c_2\end{pmatrix}\;\middle|\;c_1, c_2 \in \mathbb{R}\right\}\end{aligned}

Since \mathbf{v}_1 \text{ and } \mathbf{v}_2 are not collinear (they are not multiples of each other), they span the entire \mathbb{R}^2 plane.

Example 2: Span of Three Vectors in \mathbb{R}^3

Consider three vectors \mathbf{u}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \mathbf{u}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \text{ and } \mathbf{u}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.

The span of \{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3\} is the set of all vectors that can be written as:

c_1 \mathbf{u}_1 + c_2 \mathbf{u}_2 + c_3 \mathbf{u}_3= c_1 \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}+ c_2 \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}+ c_3 \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}

This can be written as:

\text{Span}\{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3\}= \left\{\begin{pmatrix}c_1 \\c_2 \\c_3\end{pmatrix}\;\middle|\;c_1, c_2, c_3 \in \mathbb{R}\right\}

Since \mathbf{u}_1, \mathbf{u}_2, \text{ and } \mathbf{u}_3 are linearly independent, they span the entire \mathbb{R}^3 space.

Properties of Span

Closed Under Addition and Scalar Multiplication: Any linear combination of vectors in the span of a set is also in the span. If u and v are in span{v₁, v₂, . . . ,v_n}, then c₁u + c₂v is also in the span for any scalars c₁ and c₂​.

Smallest Subspace Containing the Set: The span of a set of vectors is the smallest subspace that contains all the vectors in the set. Any subspace that contains the set must also contain the span of the set​​​​.

Redundancy and Basis: If a vector in the set can be written as a linear combination of the other vectors, it is redundant and can be removed without changing the span. The remaining set is still a spanning set. A basis is a spanning set with no redundant vectors (i.e., the vectors are linearly independent)​​​​.

Dimensionality: The dimension of the span of a set of vectors is the maximum number of linearly independent vectors in the set. This is also the number of vectors in the basis for the span​​.

Intersection with Other Subspaces: The intersection of the span of two sets of vectors is the set of all vectors that can be expressed as linear combinations of both sets. This forms a subspace itself.

Spanning Set

A set of vectors spans a space if every vector in that space can be written as a linear combination of the vectors in the set. If span{v₁, v₂, . . .,v_n} = V is a spanning set for V.

Minimal Spanning Set

A minimal spanning set, also known as a basis, is a set of vectors in a vector space that spans the entire space and is linearly independent.

Example: Find the basis of vector made from column of matrix A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}

Solution:

Form the Matrix: A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}

Row Reduce the Matrix:

\begin{aligned}& \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \\& R_2\rightarrow R_2 - 2R_1 \\& \begin{bmatrix} 1 & 3 \\ 0 & -2 \end{bmatrix} \\& R_2 \rightarrow R_2/-2 \\& \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \\& R_1\rightarrow R_1 - 3R_2 \\& \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\end{aligned}

The matrix is now in reduced row echelon form, indicating that both columns are pivot columns.

Thus, basis of the given vector are \begin{bmatrix} 1 \\ 0 \end{bmatrix} and \begin{bmatrix} 0 \\ 1 \end{bmatrix}.