Column Space Calculator

About Column Space Calculator

The Column Space Calculator finds the column space (also called the range or image) of any matrix by performing row reduction to Reduced Row Echelon Form (RREF). It identifies the pivot columns, extracts the corresponding basis vectors from the original matrix, and computes the rank and nullity. The step-by-step player shows every row operation — swaps, scaling, and elimination — so you can follow the entire process. For 2D and 3D matrices, an interactive visualization shows the column space as a line, plane, or full space.

What Is the Column Space?

The column space of a matrix A (written Col(A) or Range(A)) is the set of all linear combinations of the column vectors of A. In other words, it is the span of the columns:

$$\text{Col}(A) = \{ A\mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n \} = \text{span}(\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n)$$

The column space is a subspace of \(\mathbb{R}^m\), where m is the number of rows. Its dimension equals the rank of the matrix.

How to Find the Column Space

1. Write the matrix A — arrange your vectors as columns.
2. Row reduce to RREF — apply Gaussian elimination (row swaps, scaling, and elimination) until the matrix is in reduced row echelon form.
3. Identify pivot columns — columns that contain a leading 1 (pivot) in the RREF.
4. Extract basis from original matrix — the columns of the original matrix A at the pivot positions form a basis for the column space.

Key Concepts

Column Space vs. Row Space vs. Null Space

How to Use the Column Space Calculator

1. Set dimensions — Choose the number of rows and columns for your matrix (up to 6×6).
2. Enter values — Type numbers into each cell. Use the quick examples for preset matrices with different ranks.
3. Calculate — Click "Find Column Space" to see the full analysis.
4. Explore results — Use the step player to watch each row operation. Review the highlighted pivot columns, basis vectors, and the rank-nullity breakdown. For small matrices, check the geometric visualization.

Frequently Asked Questions

What is the column space of a matrix?

The column space of a matrix A is the set of all possible linear combinations of its column vectors. It is also called the range or image of the matrix. Geometrically, it represents all vectors that can be reached by applying the matrix transformation.

How do you find the column space of a matrix?

Row reduce the matrix to reduced row echelon form (RREF). Identify the pivot columns in the RREF. The corresponding columns from the original matrix form a basis for the column space.

What is the relationship between rank and column space?

The rank of a matrix equals the dimension of its column space. It is the number of linearly independent columns, which equals the number of pivot columns in the RREF.

What is the rank-nullity theorem?

The rank-nullity theorem states that for an m×n matrix A, rank(A) + nullity(A) = n, where n is the number of columns. The rank is the dimension of the column space and the nullity is the dimension of the null space.

Can the column space be empty?

The column space always contains at least the zero vector. If the matrix is the zero matrix, the column space is just the zero vector set. For any nonzero matrix, the column space is a nontrivial subspace.

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