Find the column space of a matrix including basis vectors, rank, nullity, and RREF steps. Supports matrices up to 6×6 with presets and step-by-step row reduction.
The column space of a matrix is the set of all possible linear combinations of its column vectors — a fundamental concept in linear algebra that determines what outputs a linear system can produce. Understanding the column space tells you whether a system of equations has a solution for a given right-hand side: if the target vector lies in the column space, a solution exists. This calculator takes any matrix up to 6×6, performs row reduction to reduced row echelon form (RREF), identifies pivot columns, and extracts a basis for the column space from the original matrix. It reports the rank (dimension of the column space), nullity (dimension of the null space), and verifies the rank-nullity theorem. You can view every row operation step-by-step, making it ideal for students learning Gaussian elimination. Presets include the identity matrix, rank-deficient matrices, and matrices with zero columns so you can explore different scenarios instantly. Engineers use column space analysis for signal processing, data compression, and least-squares fitting, while mathematicians use it to study linear transformations and subspaces.
Row-reducing a matrix to RREF by hand is a multi-step process prone to arithmetic errors, especially for matrices larger than 2×2. This calculator performs Gaussian elimination automatically, identifies pivot columns, extracts basis vectors from the original matrix, and verifies the rank-nullity theorem. The step-by-step breakdown shows every row operation so students can follow the process and check their own work. Engineers use it for quick subspace analysis in signal processing and least-squares problems.
Col(A) = span of pivot columns of A Rank(A) = number of pivot columns Rank + Nullity = number of columns
Result: Rank = 2, Nullity = 1
For A = [[1,2,3],[4,5,6],[7,8,9]], RREF reveals 2 pivot columns. Rank = 2, Nullity = 1. Basis = {(1,4,7), (2,5,8)}.
The column space Col(A) is the set of all vectors b for which the system Ax = b has a solution. If b lies in the column space, a solution exists; if not, no solution exists. The column space is spanned by the pivot columns of the original matrix — these are the linearly independent columns, and every other column can be expressed as a linear combination of them. The dimension of the column space equals the rank of the matrix.
The rank-nullity theorem states that rank(A) + nullity(A) = n, where n is the number of columns. The rank counts independent columns (dimensions of the column space), and the nullity counts the number of free variables (dimensions of the null space). This theorem provides a powerful check: if you know the rank, you immediately know the nullity, and vice versa. A matrix with full column rank has nullity 0, meaning Ax = b has at most one solution for any b.
In data science, the column space relates to Principal Component Analysis (PCA): the principal components span the most important subspace of the data. In signal processing, the column space of a mixing matrix determines which signals can be recovered. In structural engineering, the rank of a stiffness matrix determines whether a structure is statically determinate. Least-squares fitting projects a vector onto the column space to find the best approximation when an exact solution does not exist.
The column space (or range) is the set of all vectors that can be expressed as a linear combination of the matrix columns. It forms a subspace of the output space.
Row-reduce the matrix to RREF, identify pivot columns, then take the corresponding columns from the original matrix as basis vectors. Use this as a practical reminder before finalizing the result.
The rank equals the dimension of the column space — the number of linearly independent columns. A higher rank means the matrix maps to a larger subspace.
It states that rank + nullity = number of columns. Nullity is the dimension of the null space (solutions to Ax = 0).
RREF changes the column space; only the original columns preserve the actual span. RREF identifies which columns are pivots, but the basis vectors come from the original matrix.
All columns are linearly independent, the null space is {0}, and the system Ax = b has at most one solution for any b. Keep this note short and outcome-focused for reuse.