How To Find The Span Of A Set Of Vectors

Finding the span of a set of vectors is a fundamental concept in linear algebra, representing the set of all possible linear combinations of those vectors. Understanding how to find the span is crucial for grasping concepts like vector spaces, linear independence, and bases. This comprehensive guide will walk you through the process, providing a clear understanding with detailed explanations and practical examples.

Understanding the Span of a Set of Vectors

The span of a set of vectors is essentially the "reach" of those vectors when combined linearly. A linear combination involves multiplying each vector by a scalar (a real number) and then adding the resulting vectors together. The span encompasses every possible vector you can create using this process.

Formally, if you have a set of vectors v1, v2, ..., vn in a vector space V, then the span of these vectors, denoted as span(v1, v2, ..., vn), is the set of all vectors that can be written as:

c1v1 + c2v2 + ... + cnvn

where c1, c2, ..., cn are scalars.

Why is Finding the Span Important?

Understanding Vector Spaces: The span helps define and understand vector spaces. If the span of a set of vectors equals the entire vector space, it means that those vectors can generate any vector within that space.
Determining Linear Independence: The concept of the span is closely related to linear independence. If none of the vectors in the set can be written as a linear combination of the others, then the set is linearly independent, and its span has maximum "coverage."
Finding a Basis: A basis for a vector space is a set of linearly independent vectors that span the entire space. Finding the span is a crucial step in identifying a basis.
Solving Linear Systems: Spans play a significant role in understanding the solutions to systems of linear equations. The solution set often forms a span of certain vectors.

Steps to Find the Span of a Set of Vectors

The process of finding the span can vary depending on the context and the specific set of vectors. Here's a generalized step-by-step approach:

Understand the Vector Space: Identify the vector space in which the vectors reside (e.g., R², R³, or a more abstract vector space). This determines the potential "reach" of the span.
Form a Linear Combination: Write a general linear combination of the given vectors using scalar multipliers (e.g., c1, c2, c3).
Express the Linear Combination as a System of Equations: Set the linear combination equal to a general vector in the vector space. This will result in a system of linear equations.
Analyze the System of Equations: Determine whether the system of equations is consistent (i.e., has a solution) for all possible values of the components of the general vector. This usually involves row-reducing the augmented matrix.
Interpret the Results:
- If the system is consistent for all possible vectors, the span is the entire vector space.
- If the system has restrictions on the components of the general vector for it to be consistent, then the span is a subspace of the vector space, and the restrictions define the span.

Let's illustrate this with several examples.

Example 1: Finding the Span of Two Vectors in R²

Consider the set of vectors v1 = (1, 0) and v2 = (0, 1) in R².

Vector Space: The vector space is R², the set of all two-dimensional vectors.
Linear Combination: Let's form a linear combination: c1v1 + c2v2 = c1(1, 0) + c2(0, 1) = (c1, c2)
System of Equations: We want to see if this linear combination can produce any vector (x, y) in R². So, we set (c1, c2) = (x, y).
Analysis: This gives us the equations:
- c1 = x
- c2 = y
For any values of x and y, we can find corresponding values of c1 and c2 that satisfy the equations.
Interpretation: Since the system is consistent for all (x, y) in R², the span of v1 and v2 is the entire vector space R². span(v1, v2) = R². This means that v1 and v2 form a basis for R².

Example 2: Finding the Span of Two Vectors in R³

Let's consider the vectors v1 = (1, 0, 0) and v2 = (0, 1, 0) in R³.

Vector Space: R³, the set of all three-dimensional vectors.
Linear Combination: c1v1 + c2v2 = c1(1, 0, 0) + c2(0, 1, 0) = (c1, c2, 0)
System of Equations: We want to see if we can get any vector (x, y, z) in R³. So, we set (c1, c2, 0) = (x, y, z).
Analysis: This gives us the equations:
- c1 = x
- c2 = y
- 0 = z
Notice that the third equation, 0 = z, imposes a restriction. This means we can only reach vectors where the z-component is 0.
Interpretation: The span of v1 and v2 is the set of all vectors of the form (x, y, 0). This is the xy-plane in R³. span(v1, v2) = {(x, y, 0) | x, y ∈ R}. The span is a subspace of R³, specifically, the xy-plane.

Example 3: Finding the Span of a Single Vector in R³

Let's consider the vector v = (1, 2, 3) in R³.

Vector Space: R³, the set of all three-dimensional vectors.
Linear Combination: cv = c(1, 2, 3) = (c, 2c, 3c)
System of Equations: We want to see if we can get any vector (x, y, z) in R³. So, we set (c, 2c, 3c) = (x, y, z).
Analysis: This gives us the equations:
- c = x
- 2c = y
- 3c = z
From these equations, we can deduce:
- y = 2x
- z = 3x
These are restrictions on the components of the vector (x, y, z).
Interpretation: The span of v is the set of all vectors of the form (x, 2x, 3x). This is a line passing through the origin in R³, with direction vector (1, 2, 3). span(v) = {(x, 2x, 3x) | x ∈ R}.

Example 4: Finding the Span with Redundant Vectors

Consider the vectors v1 = (1, 0), v2 = (0, 1), and v3 = (1, 1) in R².

Vector Space: R², the set of all two-dimensional vectors.
Linear Combination: c1v1 + c2v2 + c3v3 = c1(1, 0) + c2(0, 1) + c3(1, 1) = (c1 + c3, c2 + c3)
System of Equations: We want to see if we can get any vector (x, y) in R². So, we set (c1 + c3, c2 + c3) = (x, y).
Analysis: This gives us the equations:
- c1 + c3 = x
- c2 + c3 = y
We can solve for c1 and c2 in terms of c3:
- c1 = x - c3
- c2 = y - c3
Since we can find solutions for c1 and c2 for any values of x and y (by choosing an arbitrary value for c3), the system is consistent for all vectors in R².
Interpretation: The span of v1, v2, and v3 is the entire vector space R². span(v1, v2, v3) = R². Notice that v3 is a linear combination of v1 and v2 (v3 = v1 + v2), so it's redundant. It doesn't add anything new to the span that wasn't already covered by v1 and v2.

Example 5: Finding the Span in a Vector Space of Polynomials

Let P2 be the vector space of polynomials of degree at most 2. Consider the polynomials p1(t) = 1 + t and p2(t) = 1 - t. Find the span of p1(t) and p2(t).

Vector Space: P2, the set of all polynomials of the form at² + bt + c, where a, b, c are real numbers.
Linear Combination: c1p1(t) + c2p2(t) = c1(1 + t) + c2(1 - t) = (c1 + c2) + (c1 - c2)t
System of Equations: We want to see if we can get any polynomial at² + bt + c in P2. So, we set (c1 + c2) + (c1 - c2)t = at² + bt + c. This is tricky because we need to consider the t² term. However, notice that c1p1(t) + c2p2(t) will never have a t² term.
Analysis: Equating coefficients, we get:
- c1 + c2 = c
- c1 - c2 = b
- a = 0
The equation a = 0 is a crucial restriction.
Interpretation: The span of p1(t) and p2(t) is the set of all polynomials of the form bt + c (i.e., all polynomials of degree at most 1). span(p1(t), p2(t)) = {bt + c | b, c ∈ R}. This is a subspace of P2. It doesn't include any quadratic polynomials.

General Techniques and Considerations

Row Reduction (Gaussian Elimination): For more complex systems of equations, row reduction is an indispensable tool to determine consistency and find the span. Express the system of equations as an augmented matrix and row-reduce it to echelon form or reduced row-echelon form. The resulting matrix will reveal any restrictions on the components of the general vector.
Linear Independence: If the vectors are linearly independent, the span will be "larger" (in terms of dimension) than if they are linearly dependent. If the vectors are linearly dependent, some of them are redundant and can be removed without changing the span.
Dimension: The dimension of the span is the number of linearly independent vectors in the set. For example, if you have three vectors in R³ and they are linearly independent, their span is R³ (dimension 3). If they are linearly dependent, their span will be a plane (dimension 2) or a line (dimension 1).
Geometric Interpretation: Visualizing the span can be helpful, especially in R² and R³. The span of a single non-zero vector is a line, the span of two linearly independent vectors is a plane, and so on.
Abstract Vector Spaces: The same principles apply to abstract vector spaces, such as the space of matrices, functions, or sequences. However, the "vectors" are now matrices, functions, or sequences, and the linear combinations involve scalar multiplication and addition of these objects.

Common Pitfalls to Avoid

Assuming the Span is Always the Entire Vector Space: The span is not always the entire vector space. It can be a subspace, and it's crucial to identify the restrictions that define that subspace.
Ignoring Linear Dependence: Failing to recognize linear dependence can lead to unnecessary complexity. Identifying and removing redundant vectors simplifies the process.
Incorrectly Setting up the System of Equations: Ensuring that the linear combination is correctly set equal to a general vector in the vector space is paramount. This step is where many errors occur.
Misinterpreting Row Reduction Results: Correctly interpreting the results of row reduction is crucial. Pay attention to any rows that imply restrictions on the components of the general vector.

FAQ: Finding the Span of a Set of Vectors

Q: What does it mean if the span of a set of vectors is the zero vector?

A: This means that all the vectors in the set are the zero vector. The only linear combination you can form is the zero vector itself.

Q: Can the span of a set of vectors be empty?

A: No, the span is never empty. It always contains at least the zero vector, since setting all scalars to zero in the linear combination will always result in the zero vector.

Q: How do I find the span of an infinite set of vectors?

A: The concept of the span extends to infinite sets of vectors. However, in practice, you often look for a finite subset of the infinite set that spans the same space. This involves identifying a basis for the span.

Q: How does the concept of span relate to column space of a matrix?

A: The column space of a matrix is the span of its column vectors. If you treat the columns of a matrix as vectors, their span forms the column space. This is a fundamental connection in linear algebra.

Q: Is the span of a set of vectors always a vector space?

A: Yes, the span of any set of vectors is always a vector space. It satisfies all the axioms of a vector space, including closure under addition and scalar multiplication.

Conclusion

Finding the span of a set of vectors is a fundamental skill in linear algebra. By understanding the underlying principles and following the step-by-step approach outlined in this guide, you can confidently determine the span of various sets of vectors in different vector spaces. Remember to pay attention to linear independence, row reduction techniques, and the geometric interpretation of the span to gain a deeper understanding. Mastering this concept will significantly enhance your ability to tackle more advanced topics in linear algebra and its applications.