Column Space Calculator

How Column Space Calculator Works?

The Column Space Calculator analyzes a 3×3 matrix assembled from the nine entered integer values. If the matrix is $A$, its columns are vectors in $R^3$. The column space is the set of every linear combination of these columns.

To identify independent columns, the calculator performs Gaussian elimination and displays the reduced row echelon form (RREF). A pivot in column $j$ means that the original column $c_j$ contributes to a basis. Row operations change the appearance of the matrix but preserve the relationships among its columns, so the basis is taken from the original matrix rather than from the RREF.

The rank is also the dimension of the column space. Because this calculator uses a 3×3 matrix, the rank can be 0, 1, 2, or 3. The nullity follows from the rank–nullity theorem and counts the free variables in the homogeneous system $Ax=0$.

When the nullity is greater than zero, the columns are linearly dependent. The calculator reports relationships obtained from the free variables in $Ax=0$, showing how non-pivot columns can be written as combinations of pivot columns.

For a square 3×3 matrix, the determinant provides an additional full-rank test:

Practical Use Cases

Students can enter textbook or homework matrices to verify row reduction, locate pivot columns, identify a valid column-space basis, and check the rank–nullity relationship.

Mathematics educators can use the calculator to demonstrate how RREF reveals pivots while the corresponding basis vectors must still be selected from the original matrix. It also provides quick examples of independent and dependent columns.

Engineers can use rank and determinant results to check whether a small system contains independent equations or whether a matrix has redundant structural directions.

Data and modeling practitioners can inspect a 3×3 set of feature or coefficient columns for linear dependence. A reduced rank indicates that some columns can be represented using others.

Researchers and analysts can use the nullity and dependency output to identify free variables and describe non-unique solutions to homogeneous linear systems represented by a 3×3 matrix.

Accuracy and Limitations

The calculator is designed for exactly nine integer inputs, with each entry restricted to the range −1000 through 1000 and an integer step of 1. It computes the 3×3 matrix, RREF, pivot columns, original-column basis, rank, column-space dimension, nullity, dependencies, and determinant from those values. Its conclusions are exact for integer arithmetic; however, it does not accept matrices larger or smaller than 3×3, decimal or symbolic entries are outside the supported input format, and near-zero numerical tolerance is not relevant because the inputs are integers. A zero determinant indicates that the matrix is not full rank, but the determinant alone does not identify which columns form the basis; those are determined from the RREF pivots.

Quick Examples