Free Variable Vector Calculator

Enter Matrix Data

Matrix entries

Use one row per line. Separate numbers with spaces or commas.

Right hand side vector

Use this only when the selected mode needs a separate vector.

System mode

Variable prefix

Decimal places

Zero tolerance

Formula Used

The calculator reduces the augmented matrix [A | b] to reduced row echelon form. Pivot columns contain leading ones. Non-pivot columns are free variables.

For each pivot row, the relation is x_p = b_p - a_p,f1t₁ - a_p,f2t₂ - ... .

The vector solution is x = x_particular + t₁v₁ + t₂v₂ + ... . Rank plus nullity equals the number of variables.

How to Use This Calculator

Enter the coefficient matrix or an augmented matrix.
Select the matching system mode.
Enter a separate right side vector only when needed.
Set decimal places and tolerance for clean rounding.
Press Calculate to show results below the header.
Use CSV or PDF buttons to save the result.

Example Data Table

Mode	Matrix Input	Expected Free Variable	Vector Form
Last matrix column is right side	1 2 -1 3 0 1 4 5	x3	x = [-7, 5, 0]^T + t1[9, -4, 1]^T
Homogeneous Ax = 0	1 2 3 2 4 6	x2, x3	x = t1[-2, 1, 0]^T + t2[-3, 0, 1]^T
Use separate right side Ax = b	1 0 2 0 1 -1	x3	Depends on the entered right side vector

Understanding Free Variable Vectors

A free variable vector describes every solution direction left after row reduction. In a linear system, some columns become pivot columns. Those variables are controlled by leading ones. Other columns stay non-pivot columns. Those variables can take any value. Each free value creates a parameter, such as t1 or t2.

Why Free Variables Matter

Free variables explain whether a system has one solution, no solution, or infinitely many solutions. When every variable is a pivot variable, the solution is fixed. When at least one variable is free, the answer becomes a family of vectors. This family is useful in linear algebra, data fitting, computer graphics, engineering models, and matrix theory.

How This Calculator Works

This calculator reduces the entered matrix to reduced row echelon form. It then finds pivot columns, free columns, matrix rank, and nullity. For a homogeneous system, the right side is zero. For a nonhomogeneous system, the calculator also builds a particular vector when the system is consistent. The final answer is written as a particular vector plus parameter vectors.

Interpreting the Vector Form

A vector form separates fixed values from free movement. The fixed vector gives one solution. Each parameter vector shows one direction that can be added without breaking the equations. If the system is homogeneous, the fixed vector is the zero vector. The parameter vectors then form a basis for the null space.

Practical Input Tips

Enter one matrix row per line. Use spaces, commas, or tabs between values. Fractions like 3/4 are accepted. Choose a small tolerance when your entries are exact. Use a larger tolerance when decimals come from measured data. Round results only after checking the row-reduction steps.

Using Results Correctly

Always check the rank and nullity. Rank counts independent pivot columns. Nullity counts free variables. Their sum equals the number of variables. If an inconsistent row appears, the system has no vector solution. If the solution is consistent, the displayed parameter vectors give a compact answer. You can export the work as CSV or PDF for homework notes, reports, or later review.

This makes long matrix work easier to audit. It helps compare, share, and reuse technical lessons. It reduces copying mistakes during review.

FAQs

What is a free variable?

A free variable is a variable without a pivot column after row reduction. It can take any value, so it becomes a parameter in the vector solution.

What is a free variable vector?

It is a direction vector attached to a parameter. Adding this vector keeps the system true because it follows the dependency shown in reduced row echelon form.

Can this calculator handle augmented matrices?

Yes. Select the mode where the last matrix column is the right side. The calculator splits that column from the coefficient matrix automatically.

Can I solve homogeneous systems?

Yes. Select homogeneous mode. The calculator uses a zero right side and returns null space vectors when free variables are present.

What does nullity mean?

Nullity is the number of free variables. It also equals the number of direction vectors in the homogeneous solution space.

What does rank mean?

Rank is the number of pivot columns. It tells how many independent variable columns remain after row reduction.

Why does the calculator show no solution?

No solution appears when a row reduces to all zero coefficients with a nonzero right side. That row creates a contradiction.

Can I enter fractions?

Yes. Fractions such as 1/2, -3/4, and 5/8 are accepted. They are converted to decimal values for the row-reduction process.