Left Null Space Calculator

Calculator

Matrix A

Use spaces, commas, or tabs between entries. Use new lines or semicolons between rows.

Zero tolerance

Decimal places

Basis display style

Calculation target

Allowed entries

Result checks

Show reduced row echelon form of A transpose

Formula Used

The left null space of a matrix A is the null space of A transpose.

Left null space: N(A^T) = { y : A^Ty = 0 }

Dimension: dim N(A^T) = m - rank(A), where m is the number of rows in A.

The calculator forms A transpose, reduces it to echelon form, finds free variables, and builds one basis vector for each free variable.

How to Use This Calculator

Enter matrix rows in the matrix box.
Separate numbers with spaces, commas, or tabs.
Set a zero tolerance for rounding small values.
Choose how basis vectors should be displayed.
Submit the form and read the result above the form.
Use CSV or PDF buttons to save the calculation.

Example Data Table

Matrix A	Rows x Columns	Expected Rank	Expected Left Nullity	Use Case
`1 2 3; 2 4 6; 1 1 1; 3 5 7`	4 x 3	2	2	Row dependency check
`1 0; 0 1; 1 1`	3 x 2	2	1	Simple teaching example
`1 2; 3 4`	2 x 2	2	0	Independent rows

Understanding the Left Null Space

The left null space is a key subspace in linear algebra. It contains every vector y that makes A transpose y equal zero. For an m by n matrix A, each left null vector has m entries. These entries weight the rows of A. When the weighted row combination becomes the zero row, the weights form a valid left null vector.

Why It Matters

This space explains hidden row dependencies. A nonzero vector in the left null space proves that some rows are not independent. That fact is useful in systems of equations, network balances, regression diagnostics, engineering constraints, and data checks. It also connects directly with rank. If A has m rows and rank r, then the left null space has dimension m minus r.

How This Calculator Works

The calculator builds A transpose and solves A transpose y equals zero. It reduces A transpose to row echelon form. Pivot columns describe dependent variables. Free columns create basis directions. For each free variable, the tool sets that free value to one, sets the other free values to zero, and back solves for pivot values. The resulting vectors form a basis.

Input Flexibility

You may enter integers, decimals, negative values, fractions, and scientific notation. Rows can be pasted from spreadsheets. Each row should contain the same number of entries. The tolerance option helps handle small roundoff values. Decimal places control display only. They do not change the internal calculation.

Reading the Result

The rank shows how many independent rows or columns exist. The left nullity shows the number of independent left null vectors. If the nullity is zero, only the zero vector satisfies the equation. If basis vectors appear, each one should pass the orthogonality check. That check multiplies A transpose by the vector and reports near zero components.

Practical Use

Use the result to identify redundant equations, check row balance models, or prepare proofs. Download the CSV for spreadsheets. Download the PDF for reports. Always review the entered matrix before using the basis in final work, especially when values are rounded or measured. For exact teaching examples, prefer rational entries, because fractions make dependencies easier to verify by hand during classroom review or audits.

FAQs

What is the left null space?

It is the set of all vectors y that satisfy A transpose times y equals zero. These vectors create zero weighted sums of the rows of A.

How is it different from the regular null space?

The regular null space solves Ax equals zero. The left null space solves A transpose y equals zero. Its vectors match the number of rows in A.

What does left nullity mean?

Left nullity is the number of independent basis vectors in the left null space. It equals the row count minus the rank of the matrix.

Can I enter fractions?

Yes. You can enter values such as 1/2, -3/4, decimals, whole numbers, and scientific notation. Separate entries with spaces, commas, or tabs.

Why does tolerance matter?

Tolerance treats very small values as zero. This helps with rounded decimal input and floating point noise, especially for nearly dependent rows.

What if no basis vector appears?

Then the left nullity is zero. Only the zero vector satisfies A transpose y equals zero within the selected tolerance.

What does the residual check show?

It multiplies A transpose by each basis vector. A valid vector should produce a zero vector or values very close to zero.

Can I use the downloads in reports?

Yes. The CSV is useful for spreadsheets. The PDF is useful for sharing the rank, nullity, basis vectors, and verification values.