Span in Linear Algebra

In linear algebra, the concept of "span" is fundamental and helps us understand how sets of vectors can generate entire spaces. The span of a set of vectors is defined as the collection of all possible linear combinations of those vectors. Essentially, if you have a set of vectors, their span includes every vector that can be formed by scaling those vectors and adding them together.

For example, if you have two vectors in a two-dimensional space, the span of these vectors can cover the entire plane if the vectors are not collinear. If they are collinear, the span will only cover a line. Similarly, in three dimensions, the span of three vectors can cover the entire space if the vectors are not coplanar.

In this article, we will discuss the concept of Span in detail including definition, example, properties as well as method to calculate the span.

What is Span in Linear Algebra?

In linear algebra, the span of a set of vectors is the set of all possible linear combinations of those vectors. The span of a set of vectors can be thought of as the "space" that the vectors occupy. For example:

• If the vectors are in two-dimensional space and they are not collinear (not multiples of each other), their span is the entire plane.
• If they are collinear, their span is just a line.
• In three-dimensional space, if three vectors are not coplanar (do not lie in the same plane), their span is the entire space. If they are coplanar, the span is a plane.

Definition of Span

If you have a set of vectors \{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}, their span is the collection of vectors that can be expressed in the form:

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

Where c₁, c₂, . . . ,c_n​ are scalars (real numbers). Essentially, the span of these vectors is the set of all vectors that can be formed by scaling and adding the original vectors.

Examples of Span

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:

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

This can be written as:

\text{Span}\{\mathbf{v}_1, \mathbf{v}_2\} = \left\{ \begin{pmatrix} a + 3b \\ 2a + 4b \end{pmatrix} \mid a, b \in \mathbb{R} \right\}

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} \mid 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

Some of the properties of span are:

• 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}.

Conclusion

The span of vectors in linear algebra is a foundational concept with numerous applications across different fields, including solving systems of linear equations, computer graphics, engineering, physics, data science, signal processing, and control theory. By understanding the span, one can gain insights into the structure and behavior of complex systems, leading to more effective solutions and innovations in various domains.

Read More,

• Linear Algebra
• Components of Linear Algebra
• Linear Independence of Vectors
• System of Linear Equations
• Matrices
• Orthogonal and Orthonormal Vectors