How To Find The Span Of A Set Of Vectors

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. Understanding how to do this is crucial for many applications, from solving systems of equations to understanding the geometry of vector spaces. This article will guide you through the process, explaining the concept and providing practical examples. We'll cover both the theoretical understanding and the practical calculation methods.

What is the Span of a Set of Vectors?

The span of a set of vectors is the set of all possible linear combinations of those vectors. In simpler terms, it's all the vectors you can create by multiplying each vector in the set by a scalar (a number) and then adding the results together. This forms a subspace of the vector space the original vectors belong to. The span essentially describes the "reach" or extent of these vectors. Think of it as the entire space that these vectors can "cover" through linear combinations.

Methods for Finding the Span

There are several ways to determine the span of a set of vectors, depending on the context and the number of vectors involved.

1. Geometric Interpretation (for 2D and 3D vectors):

For vectors in two or three dimensions, visualizing the span can be helpful.

• One Vector: The span of a single non-zero vector is a line passing through the origin and extending infinitely in both directions.
• Two Vectors: If the two vectors are linearly independent (not multiples of each other), their span is a plane passing through the origin. If they are linearly dependent, their span is a line.
• Three Vectors: In 3D space, three linearly independent vectors span the entire 3D space. If they are linearly dependent, their span could be a plane or a line, depending on the relationships between the vectors.

This geometric approach is intuitive but limited to low-dimensional spaces.

2. Using Row Reduction (Gaussian Elimination):

This is a powerful algebraic method that works for vectors of any dimension. The process involves arranging the vectors as rows in a matrix and then performing Gaussian elimination (row reduction) to obtain the row echelon form.

• Form a Matrix: Arrange the vectors as rows in a matrix.
• Perform Row Reduction: Use elementary row operations (swapping rows, multiplying a row by a non-zero scalar, adding a multiple of one row to another) to transform the matrix into row echelon form or reduced row echelon form.
• Interpret the Result: The non-zero rows in the row echelon form represent a basis for the span of the original vectors. The span is then the set of all linear combinations of these basis vectors.

Example:

Let's find the span of the vectors v1 = [1, 2, 3] and v2 = [4, 5, 6]

1. Form the Matrix:

   [ 1  2  3 ]
   [ 4  5  6 ]
   

2. Row Reduction: Subtract 4 times the first row from the second row:

   [ 1  2  3 ]
   [ 0 -3 -6 ]
   

   Then, divide the second row by -3:

   [ 1  2  3 ]
   [ 0  1  2 ]
   

   Both rows are linearly independent.

3. Interpretation: The span is the set of all vectors of the form a[1, 2, 3] + b[0, 1, 2], where a and b are scalars. This represents a plane in 3D space.

3. Linear Dependence and Independence:

The concept of linear dependence is intimately connected to finding the span. If a set of vectors is linearly dependent (one vector can be expressed as a linear combination of the others), then the span is unchanged if we remove the dependent vector. Determining linear independence often involves solving a homogeneous system of linear equations or calculating the determinant of a matrix formed by the vectors.

Conclusion:

Finding the span of a set of vectors is a fundamental skill in linear algebra. Whether you use a geometric approach (for low-dimensional vectors) or the more general method of row reduction, understanding the concept and the techniques for calculation is essential for mastering linear algebra and its numerous applications in various fields. Remember to always consider linear dependence when simplifying your calculations and determining the basis for the span.