Find Span Of Matrix Calculator

Calculate the Span of Vectors

Enter up to three vectors in 3D space (R³). For fewer vectors or lower dimensions, set unused components or vectors to zero.

Matrix Transformation

Step	Matrix
Initial	–
Row Echelon	–

Comparison of Original Vectors and Basis Vectors

What is the Span of Vectors?

In linear algebra, the span of a set of vectors is the set of all possible linear combinations of those vectors. Imagine you have a few vectors; the span is the entire space (like a line, a plane, or a 3D space) that you can reach by stretching, shrinking, and adding those original vectors together. Our span of vectors calculator helps you visualize and determine this space.

The span of a set of vectors {v1, v2, …, vk} is denoted as Span(v1, v2, …, vk) and is defined as:

Span(v1, v2, …, vk) = {c1*v1 + c2*v2 + … + ck*vk | c1, c2, …, ck are scalars}

A crucial concept related to the span is the basis of the span. A basis is a minimal set of linearly independent vectors that still span the same space. The number of vectors in the basis is called the dimension of the span. Our span of vectors calculator finds both a basis for the span and its dimension.

This calculator is useful for students learning linear algebra, engineers, physicists, and anyone working with vector spaces who needs to understand the space generated by a set of vectors.

Common misconceptions include thinking that the span is just the vectors themselves, or that the number of vectors in the original set is always the dimension of the span (which is only true if the vectors are linearly independent).

Span of Vectors Formula and Mathematical Explanation

To find the span (specifically, a basis for the span and its dimension) of a set of vectors, we typically follow these steps:

1. Form a Matrix: Create a matrix where the columns (or rows) are the vectors you are given. Let's say we have vectors v1, v2, v3 in R³. We form matrix A = [v1 v2 v3].
2. Row Reduction: Perform Gaussian elimination (row reduction) on matrix A to transform it into its Row Echelon Form (REF) or Reduced Row Echelon Form (RREF).
3. Identify Pivot Columns: In the REF or RREF, identify the columns that contain a pivot (the leading non-zero entry in a row).
4. Find the Basis: The original vectors corresponding to the pivot columns in the REF/RREF form a basis for the span of the original set of vectors. These vectors are linearly independent and span the same space.
5. Determine Dimension: The number of pivot columns (or the number of vectors in the basis) is the dimension of the span, which is also the rank of the matrix A.

Our span of vectors calculator automates this process of row reduction and pivot identification.

Variables Table

Variable	Meaning	Unit	Typical Range
v1, v2, v3…	Input vectors	Vector components (e.g., real numbers)	Any real numbers
A	Matrix formed by vectors	Matrix entries	Depends on vector components
REF/RREF	Row Echelon Form/Reduced Row Echelon Form of A	Matrix entries	Contains 0s and 1s as pivots
Basis Vectors	Linearly independent vectors spanning the space	Vector components	Subset of original vectors
Dimension	Number of vectors in the basis (rank of A)	Integer	0 to number of original vectors

Practical Examples (Real-World Use Cases)

Example 1: Linearly Dependent Vectors

Suppose we have three vectors in R²: v1 = (1, 2), v2 = (2, 4), and v3 = (3, 1). We want to find the span.

The matrix would be (in R2, so 2 rows):

A = [[1, 2, 3], [2, 4, 1]]

Row reducing this matrix:

[[1, 2, 3], [0, 0, -5]] -> [[1, 2, 0], [0, 0, 1]] (RREF)

Pivots are in columns 1 and 3. So, the original vectors v1 = (1, 2) and v3 = (3, 1) form a basis for the span. The dimension is 2. The span is R².

Using the span of vectors calculator with v1=(1,2,0), v2=(2,4,0), v3=(3,1,0) (treating as R3 with z=0), you'd find a basis like {(1,2,0), (3,1,0)} and dimension 2.

Example 2: Spanning a Line

Consider vectors v1 = (1, 1, 1) and v2 = (2, 2, 2) in R³.

Matrix A = [[1, 2], [1, 2], [1, 2]]

Row reducing: [[1, 2], [0, 0], [0, 0]]

Only one pivot column (the first). So, the basis is just {v1} = {(1, 1, 1)}. The dimension is 1. The span is a line passing through the origin in the direction of (1, 1, 1). Vector v2 is linearly dependent on v1 (v2 = 2*v1).

Our span of vectors calculator would show a basis {(1,1,1)} and dimension 1 for these inputs.

How to Use This Span of Vectors Calculator

1. Enter Vector Components: Input the x, y, and z components for each of the three vectors (v1, v2, v3). If you have fewer than three vectors, or vectors in R², set the unnecessary components or vectors to zero (e.g., for v1=(1,2) in R², enter v1x=1, v1y=2, v1z=0, and set v2 and v3 to (0,0,0) if not used).
2. Calculate: Click the "Calculate Span" button. The calculator will automatically perform the calculations.
3. View Primary Result: The "Basis will be shown here" section will display the set of vectors that form a basis for the span.
4. Examine Intermediate Results: See the original matrix formed, its row echelon form, and the calculated dimension of the span (rank).
5. Matrix Transformation Table: The table shows the initial matrix and its row echelon form for clarity.
6. Dimension Chart: The chart visually compares the number of original vectors you input (non-zero ones implicitly) and the dimension of the space they span (number of basis vectors).
7. Reset or Copy: Use "Reset" to clear inputs or "Copy Results" to copy the findings.

The results from the span of vectors calculator tell you the fundamental, independent directions that generate your vector space and how many such directions there are.

Key Factors That Affect Span Results

1. Linear Dependence/Independence: If the input vectors are linearly independent, they will all be part of the basis, and the dimension will equal the number of vectors (up to the dimension of the ambient space). If they are dependent, the basis will have fewer vectors.
2. Number of Vectors: More vectors can potentially span a larger space, but only if they add new independent directions.
3. Dimension of the Ambient Space: If you input vectors in R³, the dimension of their span cannot exceed 3.
4. Zero Vectors: Adding zero vectors does not change the span.
5. Scalar Multiples: If one vector is a scalar multiple of another, it's linearly dependent and won't be needed in the basis if the other is included.
6. Component Values: The specific numerical values of the vector components determine their relationships and the resulting span.

Understanding these factors helps interpret the output of the span of vectors calculator.

Frequently Asked Questions (FAQ)

What is the span of a single vector?

The span of a single non-zero vector is the line passing through the origin in the direction of that vector. If the vector is the zero vector, its span is just the origin itself (dimension 0).

What is the span of two vectors?

If the two vectors are linearly independent (not scalar multiples of each other), their span is a plane containing the origin and both vectors. If they are linearly dependent, their span is a line (if at least one is non-zero) or the origin (if both are zero).

Can the dimension of the span be greater than the number of vectors?

No, the dimension of the span is always less than or equal to the number of vectors in the original set.

Can the dimension of the span be greater than the dimension of the space the vectors live in (e.g., >3 for vectors in R³)?

No, the dimension of the span cannot exceed the dimension of the ambient space (e.g., 3 for R³).

Is the basis for the span unique?

No, a given vector space or subspace (like the span) can have infinitely many different bases. However, all bases for the same space will have the same number of vectors (the dimension). Our span of vectors calculator provides one such basis derived from the original vectors.

What does it mean if the dimension of the span is 0?

It means the only vector that can be formed is the zero vector, which happens if all input vectors are the zero vector.

How is the span related to the column space of a matrix?

If you form a matrix with the vectors as columns, the span of those vectors is exactly the column space of the matrix. The dimension of the span is the rank of the matrix.

What if I have more than 3 vectors or vectors in more than 3 dimensions?

This calculator is designed for up to 3 vectors in R³. For more vectors or higher dimensions, the same principles of row reduction apply, but the matrix would be larger and the calculations more extensive, often requiring software like MATLAB, Python with NumPy, or more advanced calculators.

Related Tools and Internal Resources

• Matrix Rank Calculator: Find the rank of a matrix, which is the dimension of the column or row space (and the span of its columns/rows).
• Linear Independence Calculator: Determine if a set of vectors is linearly independent or dependent.
• Eigenvalue and Eigenvector Calculator: Calculate eigenvalues and eigenvectors for a given matrix.
• Gaussian Elimination Calculator: See the step-by-step row reduction of a matrix.
• Vector Addition Calculator: Add vectors together component-wise.
• Dot Product Calculator: Calculate the dot product of two vectors.