Column Space Calculator – Matrix Column Space
Calculate the column space and basis vectors of a matrix
Table of Contents
How to Use
- Enter your matrix with rows separated by new lines
- Separate values in each row with spaces
- Click calculate to find the column space
- Review the basis vectors and rank
What is Column Space?
The column space (or range) of a matrix A is the set of all possible linear combinations of its column vectors. It represents all vectors that can be obtained by multiplying the matrix by any vector.
If A is an m×n matrix, the column space is a subspace of ℝᵐ. The dimension of this subspace is called the rank of the matrix.
Finding a Basis for Column Space
To find a basis for the column space:
- Perform row reduction on the matrix to find the pivot columns
- The pivot columns in the original matrix form a basis for the column space
- The number of pivot columns equals the rank of the matrix
Key Properties
- The rank equals the dimension of the column space
- The column space contains the zero vector
- Row operations do not change the column space structure
- The column space is the orthogonal complement of the left null space
Frequently Asked Questions
- What is the difference between column space and row space?
- Column space is the span of column vectors (subspace of ℝᵐ), while row space is the span of row vectors (subspace of ℝⁿ). Both have the same dimension (the rank), but exist in different spaces.
- How does rank relate to column space?
- The rank of a matrix equals the dimension of its column space. It represents the maximum number of linearly independent column vectors in the matrix.
- Can the column space be empty?
- No, the column space always contains at least the zero vector, making it a valid subspace. Even a zero matrix has a column space (containing only the zero vector).