Basis for Subspace Calculator

Calculator

Calculation Type

Vector Direction

Delimiter

Pivot Tolerance

Decimal Precision

Supported Values

Integers, decimals, fractions, and negatives

Matrix or Vector List

Example Data Table

Vector Number	Entered Vector	Expected Role	Reason
1	[1, 2, 3]	Basis vector	First pivot direction
2	[2, 4, 6]	Dependent	Twice vector 1
3	[0, 1, 1]	Basis vector	Adds a new direction
4	[1, 3, 4]	Dependent	Vector 1 plus vector 3

Formula Used

The calculator uses row reduction to change a matrix into reduced row echelon form.

RREF(A) shows pivot positions. A pivot column contains a leading one after reduction.

Rank(A) = number of pivot columns.

Dimension of span = rank(A).

For span mode, vectors are placed as matrix columns. Pivot columns identify independent original vectors.

For row space, the nonzero rows of the reduced matrix form a basis.

For null space, free variables create special solution vectors. Pivot variables follow this rule:

x_pivot = -RREF entry linked to the free variable.

How to Use This Calculator

Select the calculation type.
Choose whether your vectors are written as rows or columns.
Enter one row per line in the input box.
Use spaces, commas, or semicolons between values.
Set tolerance for decimal data.
Set display precision for cleaner answers.
Press the calculate button.
Review basis vectors, rank, dimension, and row steps.
Download CSV or PDF after the result appears.

Understanding a Basis for a Subspace

A basis explains a subspace with the smallest useful list of vectors. The vectors must span the same set. They must also be linearly independent. This calculator helps you test both ideas quickly.

Why Basis Matters

Many students start with a long spanning set. Some vectors repeat earlier direction. Some vectors add no new dimension. Row reduction reveals which vectors matter. Pivot positions show the independent structure. The rank tells how many dimensions the subspace has.

Supported Subspace Options

The tool accepts rows, columns, or a standard matrix. You can find a basis for a span. You can also study column space, row space, or null space. This makes the page useful for algebra homework, engineering models, graphics, data science, and systems analysis.

Span Mode

For span mode, enter each vector on a new line. Choose whether vectors are listed by rows or columns. The calculator builds the correct matrix and finds pivot columns. Those pivot columns identify a clean basis from the original vectors. This keeps the answer easy to compare with class notes.

Matrix Spaces

Column space mode uses the original pivot columns of the matrix. Row space mode uses the nonzero rows of the reduced matrix. Null space mode builds special solution vectors from free variables. Each mode reports rank, dimension, pivot columns, and dependence information.

Input Accuracy

The tolerance box helps with decimal data. A small tolerance treats tiny roundoff errors as zero. Precision controls the displayed decimals. Fraction values, such as 3/5, are accepted. This helps when exact looking input is easier than decimals.

Interpreting Results

A good basis is compact. It removes redundant vectors without changing the subspace. It also gives a coordinate system inside that subspace. Every vector in the span can be created from a unique combination of basis vectors. That is why basis, rank, and dimension are central ideas in linear algebra.

Exporting Work

Use the export buttons after calculation. The CSV file is helpful for spreadsheets. The PDF file is useful for reports or saved solutions. Always check that every row has the same number of entries before solving. Clean input gives clean results.

Review Benefit

Because the result is built from elimination, it is transparent. You can follow every pivot, compare original vectors, and explain why removed vectors were dependent during review or later exam practice.

FAQs

What is a basis for a subspace?

A basis is a set of vectors that spans the subspace and stays linearly independent. It gives the smallest complete vector list needed to describe that subspace.

What does rank mean here?

Rank is the number of pivot positions found during row reduction. It equals the dimension of the column space and row space.

Can I enter fractions?

Yes. You can enter values such as 1/2, -3/4, or 5/6. The calculator converts them into decimal values for reduction.

Should vectors be rows or columns?

Use rows when each line is one vector. Use columns when each column of the matrix is one vector. The selection matters in span mode.

Why are some vectors marked dependent?

A dependent vector can be made from earlier pivot vectors. It does not add a new direction, so it is removed from the basis.

What is the null space basis?

The null space basis contains vectors that solve Ax = 0. Each free variable creates one special solution vector.

What tolerance should I use?

Use a small value such as 0.000000001 for most decimal inputs. Increase it only when tiny rounding errors create unwanted pivots.

Can I export the result?

Yes. After calculation, use the CSV or PDF buttons. They save rank, dimension, basis vectors, pivot data, and the reduced matrix.