Calculator Input
Example Data Table
| Example matrix A | Target vector b | Rank | Nullity | Target in Col(A) | One solution x |
|---|---|---|---|---|---|
| [1 2 0; 0 1 1; 2 5 1] | [3, 2, 8] | 2 | 1 | Yes | [1, 1, 1] |
This sample shows a dependent square matrix with a one-dimensional null space and a target vector that still lies inside the column space.
Formula Used
1) Rank from pivots
Reduce matrix A to reduced row echelon form. The number of pivot columns equals the rank: rank(A) = number of pivots in RREF(A).
2) Nullity from the rank-nullity theorem
If A has n columns, then: nullity(A) = n − rank(A). This gives the dimension of the solution space of Ax = 0.
3) Column-space membership test
A target vector b belongs to the column space of A exactly when the augmented system is consistent: b ∈ Col(A) if and only if Ax = b has at least one solution.
4) Basis extraction rules
Pivot columns from the original matrix form a basis for the column space. Nonzero rows of RREF form a basis for the row space. Free-variable vectors built from RREF form a basis for the null space.
5) Determinant for square matrices
For square matrices, elimination reveals whether the matrix is invertible. A nonzero determinant means the columns are linearly independent and the matrix has full rank.
How to Use This Calculator
- Enter the number of matrix rows and columns.
- Type the matrix, one row per line, using spaces or commas between values.
- Optionally enter a target vector with the same number of entries as the matrix has rows.
- Set a small pivot tolerance if you expect decimal rounding noise.
- Press Analyze Matrix Vector Space.
- Read the result section above the form for rank, nullity, bases, RREF, and target-vector consistency.
- Use the CSV button to export tabular results.
- Use the PDF button to save the current report as a portable document.
FAQs
1) What does this calculator actually measure?
It studies the vector-space structure generated by a matrix. It reports rank, nullity, bases for key subspaces, and whether a supplied target vector can be produced from the matrix columns.
2) Why is RREF important here?
RREF makes pivots easy to see. Those pivots reveal rank, independent columns, free variables, and whether the augmented system is consistent when testing a target vector.
3) What is the difference between column space and row space?
The column space contains all linear combinations of the matrix columns. The row space contains all linear combinations of the rows. Both spaces share the same dimension, equal to rank.
4) What does nullity tell me?
Nullity tells you how many independent directions solve Ax = 0. A larger nullity means more free variables and more dependence among the matrix columns.
5) Why can a square matrix still fail to be invertible?
A square matrix is not invertible when its columns are dependent. In that case, rank is below the matrix size and the determinant becomes zero.
6) What happens if my target vector is not in the column space?
The augmented matrix becomes inconsistent after row reduction. The calculator marks the target as outside the column space and does not report a valid coefficient solution.
7) Can I use decimals and negative numbers?
Yes. The parser accepts integers, decimals, and negative values. The tolerance field helps control how very small numbers are treated during elimination.
8) Why are basis vectors shown as rows in some tables?
They are displayed as rows only for readability in the report. Conceptually, each listed row is one basis vector for the stated subspace.