Vector Space Dimension Calculator

Calculator Input

Enter vectors as matrix columns. The number of rows is the ambient dimension. The number of columns is the total vector count.

Ambient Dimension (Rows)

Number of Vectors (Columns)

Display Precision

Vector 1 · Entry 1

Vector 2 · Entry 1

Vector 3 · Entry 1

Vector 4 · Entry 1

Vector 1 · Entry 2

Vector 2 · Entry 2

Vector 3 · Entry 2

Vector 4 · Entry 2

Vector 1 · Entry 3

Vector 2 · Entry 3

Vector 3 · Entry 3

Vector 4 · Entry 3

Example Data Table

This example uses four vectors in a three-dimensional space.

Coordinate	v1	v2	v3	v4
x1	1	0	1	2
x2	0	1	1	1
x3	1	1	2	3

For this set, the span dimension is 2 because only two columns remain independent after row reduction.

Formula Used

Dimension of span = rank(A)

Place the vectors as columns of a matrix A. Reduce A to reduced row echelon form. Count pivot columns. That count is the rank and also the dimension of the subspace spanned by the vectors.

Nullity = number of columns − rank(A)

Nullity measures the number of free variables in the homogeneous system A x = 0. The calculator also checks the rank-nullity identity:

rank(A) + nullity(A) = number of columns

If rank equals the number of vectors, the set is linearly independent. If rank equals the ambient dimension, the set spans the whole ambient space.

How to Use This Calculator

Choose the ambient dimension as the row count.
Choose how many vectors you want to test.
Enter each vector as one matrix column.
Use decimals or fractions like 3/2.
Click Update Grid after changing dimensions.
Click Calculate Dimension to compute results.
Review span dimension, nullity, and basis candidates.
Use the export buttons to save the report.

Frequently Asked Questions

1) What does dimension mean in a vector space?

Dimension is the number of vectors in any basis of the space. It is the largest number of linearly independent vectors that still lie inside that space.

2) Why does rank equal the span dimension?

When vectors are written as matrix columns, each pivot column represents one independent direction. Counting those pivot columns gives the rank, which matches the dimension of the span.

3) What is nullity?

Nullity counts the number of free variables in the system A x = 0. It tells you how many independent solution directions exist inside the null space.

4) Can this calculator test linear independence?

Yes. If the rank equals the number of entered vectors, then every vector contributes a new direction. That means the set is linearly independent.

5) Can the set span the whole ambient space?

Yes. If the rank equals the ambient dimension, the entered vectors span the whole surrounding space. The summary section reports this condition directly.

6) Do fractions work in the matrix inputs?

Yes. You can type values like 1/2, -3/4, 2, or 4.75. Fractions are converted to decimals during calculation.

7) Why are pivot columns useful?

Pivot columns identify which original vectors can be kept as basis vectors. They show a minimal independent subset that still spans the same subspace.

8) What happens if some vectors repeat or depend on others?

The rank will not increase for repeated or dependent vectors. The calculator will show fewer pivot columns, a smaller span dimension, and a larger dependent count.