Column Space Calculator

What is the Column Space of a Matrix?

The column space of a matrix A (written as Col(A) or Range(A)) is the set of all possible linear combinations of its column vectors. In other words, it is the span of the columns of A. The column space is a subspace of R^m, where m is the number of rows in the matrix. Its dimension equals the rank of the matrix, which is the number of linearly independent columns.

How to Find the Column Space

To find the column space of a matrix, follow these steps:

Write the matrix A - Arrange your vectors as columns of the matrix.
Row reduce to RREF - Apply Gaussian elimination (row swaps, scaling, and elimination) until the matrix is in reduced row echelon form.
Identify pivot columns - Columns that contain a leading 1 (pivot) in the RREF are the pivot columns.
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

Rank: The dimension of the column space equals the number of pivot columns, which is the rank of the matrix.

Nullity: The dimension of the null space equals the number of free variables (non-pivot columns).

Rank-Nullity Theorem: For an m x n matrix, rank(A) + nullity(A) = n, where n is the number of columns.

Pivot Column: A column containing a leading 1 in the reduced row echelon form.

To perform other matrix operations, try our Matrix Multiply Calculator, Matrix Inverse Calculator, or Dot Product Calculator.

Column Space vs. Row Space vs. Null Space

The column space (span of columns, dimension = rank, lives in R^m) differs from the row space (span of rows, dimension = rank, lives in R^n) and the null space (solutions to Ax = 0, dimension = nullity, lives in R^n). Understanding these subspaces is fundamental to linear algebra.

How to Use the Column Space Calculator

Use the grid to enter your matrix values. Adjust the number of rows and columns using the input fields. Click the example buttons to load preset matrices. The calculator automatically computes the RREF, identifies pivot columns, and displays the basis vectors for the column space along with the rank and nullity.

Applications of Column Space

The column space is used in solving linear systems, computer graphics, data compression (PCA), machine learning, and many other fields. It helps determine whether a system Ax = b has a solution and characterizes the range of a linear transformation.

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 x 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.

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