Calculator Form
Formula Used
The calculator uses Gauss-Jordan elimination to convert matrix A into reduced row echelon form.
Column space basis: original columns of A that match pivot columns in RREF(A).
Row space basis: nonzero rows of RREF(A).
Null space basis: solve Ax = 0 by assigning one free variable at a time.
Rank: number of pivots. Nullity: number of columns minus rank.
How to Use This Calculator
- Enter your matrix in the text area.
- Choose whether your vectors are stored in rows or columns.
- Select the basis type you want to display.
- Adjust decimal places and tolerance if needed.
- Click the calculate button.
- Use CSV or PDF buttons to save the result.
Example Data Table
| Example | Input matrix | Expected idea |
|---|---|---|
| Dependent columns | 1 2 3; 2 4 6; 1 1 0 | One column depends on the others. |
| Independent standard vectors | 1 0 0; 0 1 0; 0 0 1 | Rank is 3. |
| Null space case | 1 2 3; 2 4 6 | Free variables create null basis vectors. |
What Is a Basis of a Subspace?
A basis is a clean description of a subspace. It uses the fewest vectors needed to build every vector in that subspace. The vectors must also be linearly independent. No basis vector can be made from the others.
Why This Calculator Helps
This calculator reduces a matrix to row echelon form. It then finds pivot columns and free variables. Those details show rank, dimension, and useful basis vectors. The tool can study vectors written as rows or columns. It can also show row space, column space, and null space.
How the Work Is Done
The main step is Gauss-Jordan elimination. Each pivot is scaled to one. Entries above and below that pivot are removed. The final reduced matrix shows which columns are independent. Original pivot columns form a basis for the column space. Nonzero rows in the reduced matrix form a basis for the row space.
Null Space Meaning
The null space contains all solutions of Ax = 0. Pivot variables depend on free variables. Each free variable creates one special solution vector. These vectors form a basis for the null space. If there are no free variables, the null space has only the zero vector.
Rank and Dimension
The rank equals the number of pivot columns. It is also the dimension of the row space and column space. Nullity equals the number of columns minus the rank. Rank and nullity help explain how many directions the matrix keeps or loses.
Practical Uses
Use this tool for linear algebra homework, engineering models, data checks, and span problems. It helps verify independence before building a model. It also explains whether vectors span a plane, a line, or a higher dimensional subspace. The step table makes the result easier to audit and reuse.
Reading the Answer
Start with the pivot list. It tells which vectors matter most. Then compare rank with the number of vectors. Equal values mean independence. A smaller rank means some vectors are repeated combinations. Use the basis table for final copying. Use CSV for spreadsheets. Use PDF for reports or class notes. Keep exact values when fractions are important. Check rounded decimals before submitting formal answers to teachers or clients each time.
FAQs
What is a basis of a subspace?
A basis is a set of independent vectors that spans the whole subspace. Every vector in the subspace can be made from those basis vectors.
What does rank mean?
Rank is the number of pivot columns. It is also the dimension of the column space and row space.
What does nullity mean?
Nullity is the number of free variables in Ax = 0. It equals the number of columns minus the rank.
Why are pivot columns important?
Pivot columns identify independent directions. The matching original columns form a basis for the column space.
Can I enter fractions?
Yes. You can enter values like 1/2, -3/4, decimals, or whole numbers. Keep rows the same length.
What is the row space basis?
The row space basis is made from the nonzero rows of the reduced row echelon form of the matrix.
What if the null space basis is empty?
An empty null space basis means Ax = 0 has only the zero solution. There are no free variable directions.
Why use tolerance?
Tolerance treats very small numbers as zero. This helps with decimal input and reduces rounding noise during elimination.