Matrix Domain and Codomain Calculator

Calculator Inputs

Rows m

Columns n

Number setting

Matrix symbol

Domain label

Codomain label

Zero tolerance

Optional input vector x

Supported separators

Use spaces or commas inside rows. Use new lines or semicolons between rows.

Matrix entries

Formula Used

For a matrix with m rows and n columns, the standard linear transformation is:

A: Fⁿ → F^m, x ↦ Ax

The domain is Fⁿ. The codomain is F^m. Rank is the number of pivot columns after row reduction.

rank(A) + nullity(A) = n

The map is injective when rank(A) = n. It is surjective when rank(A) = m. It is bijective when both conditions are true.

How to Use This Calculator

Enter the number of matrix rows. This becomes the codomain dimension.
Enter the number of matrix columns. This becomes the domain dimension.
Type each matrix row on a new line. Separate values with spaces or commas.
Add an optional input vector if you want to compute Ax.
Press Calculate to see domain, codomain, rank, nullity, and map behavior.
Use the export buttons to save the result as a CSV or PDF file.

Example Data Table

Matrix	Size	Domain	Codomain	Likely Focus
[1 0; 0 1; 2 -1]	3 × 2	F²	F³	Injection check
[1 2 3; 0 1 4]	2 × 3	F³	F²	Surjection check
[1 2; 3 6]	2 × 2	F²	F²	Rank deficiency
[2 0; 0 5]	2 × 2	F²	F²	Invertibility check

Matrix Domain and Codomain Guide

A matrix can describe a linear transformation. It takes an input vector from one space. It returns an output vector in another space. The input space is the domain. The target space is the codomain. For an m by n matrix, the usual mapping is from R^n to R^m. The n columns define how many input coordinates are required. The m rows define how many output coordinates are produced.

Why These Spaces Matter

Domain and codomain help explain what a matrix is allowed to accept and return. They also support deeper questions. A map is injective when different inputs do not collapse to the same output. A map is surjective when every vector in the codomain can be reached. Rank, nullity, pivot columns, and dimensions tell the story.

Rank and Nullity

Rank is the dimension of the column space. It counts independent output directions. Nullity is the dimension of the null space. It counts free input directions that map to zero. The rank nullity theorem connects them. For a matrix with n columns, rank plus nullity equals n. This calculator uses row reduction to estimate these values and then explains the mapping status.

Practical Interpretation

When rank equals the number of columns, the transformation is one to one. When rank equals the number of rows, it covers the whole codomain. A square matrix with full rank is both injective and surjective. Then the map is invertible. If rank is smaller, some information is lost, or some target vectors cannot be reached.

Use in Study and Work

This tool is useful for algebra classes, engineering models, data transformations, graphics pipelines, and systems analysis. You can enter a matrix, choose labels, test a vector, and export the result. The example table gives quick patterns for common matrix sizes. Always review numerical answers when entries are rounded, very large, or very close to zero.

Good Entry Habits

Use rows for output equations and columns for input variables. Separate entries with spaces or commas. Keep dimensions small enough to inspect. Try exact integers first. Then test decimals if needed. Compare the vector result with the row formulas. This makes errors easier to find before downloading reports or sharing files.

FAQs

What is the domain of a matrix transformation?

The domain is the input vector space. For an m by n matrix, the domain is Fⁿ because the matrix needs n input coordinates.

What is the codomain of a matrix transformation?

The codomain is the declared output vector space. For an m by n matrix, the codomain is F^m because multiplication returns m output entries.

Does the calculator find the range?

It gives rank and pivot information. These describe the image dimension. The range is the actual set of reachable outputs inside the codomain.

When is a matrix map injective?

A matrix map is injective when its rank equals its number of columns. Then the nullity is zero, and no nonzero input maps to zero.

When is a matrix map surjective?

A matrix map is surjective when its rank equals its number of rows. Then every vector in the codomain can be reached by some input.

Can a non-square matrix be bijective?

No. A bijective matrix transformation between finite spaces needs equal domain and codomain dimensions. That requires a square full-rank matrix.

What does nullity mean?

Nullity is the number of free input directions that map to the zero vector. It equals the number of columns minus the rank.

Can I export the calculation?

Yes. After calculation, use the CSV or PDF button. The export includes dimensions, rank, nullity, mapping status, and row reduction output.