Domain and Codomain of a Matrix Calculator

Matrix Calculator

Map name

Rows m

Columns n

Vector space

Entry style

Interpretation

Matrix values

Enter one matrix row per line. Separate entries with spaces or commas. Semicolons also start new rows.

Example Data Table

Matrix A	Size	Linear map	Domain	Codomain	Rank	Nullity	Meaning
[1 0; 0 1; 2 -1]	3 × 2	T: R² → R³	R²	R³	2	0	Injective, not onto R³
[1 2 3; 2 4 6]	2 × 3	T: R³ → R²	R³	R²	1	2	Not injective, not onto R²
[1 2; 3 4]	2 × 2	T: R² → R²	R²	R²	2	0	Invertible linear map

Formula Used

For an m × n matrix A, the associated linear transformation is written as T(x) = Ax.

A ∈ R^m×n maps input vectors from Rⁿ into output vectors in R^m.

Domain = Rⁿ, where n is the number of columns.

Codomain = R^m, where m is the number of rows.

Rank(A) = dimension of the column space, also called the range or image.

Nullity(A) = n − Rank(A), by the rank nullity theorem.

The map is injective when Rank(A) = n. It is surjective onto the full codomain when Rank(A) = m.

How to Use This Calculator

Enter the number of rows in the matrix.
Enter the number of columns in the matrix.
Choose the vector space notation.
Type each matrix row on a separate line.
Use spaces, commas, or fractions for entries.
Press the calculate button.
Read the domain, codomain, rank, nullity, and classification.
Download the result as CSV or PDF when needed.

Understanding Matrix Domains

A matrix can describe a linear map. Its domain and codomain come from its size. When a matrix has m rows and n columns, it maps vectors with n entries into vectors with m entries. So the domain is R n, and the codomain is R m. This rule is simple, but it drives many deeper facts.

Why Dimensions Matter

Columns tell how many input coordinates are accepted. Rows tell how many output coordinates are produced. A 3 by 2 matrix accepts two dimensional input. It returns a three dimensional output. The calculator follows this convention for every rectangular matrix. It also checks whether the map loses information, fills the target space, or both.

Rank, Nullity, and Image

The rank is the number of independent columns. It is also the dimension of the image, or range. The nullity is the dimension of the kernel. It tells how many independent input directions collapse to the zero vector. Rank and nullity always add to the number of columns. This is the rank nullity theorem.

Injective and Surjective Checks

A map is injective when different inputs always produce different outputs. For matrices, this happens when rank equals the number of columns. A map is surjective onto the full codomain when every target vector can be reached. This happens when rank equals the number of rows. A square matrix is invertible when both conditions hold.

Using the Results Carefully

The calculator assumes real vector spaces. It reads the matrix as A in the formula T(x) equals Ax. Fractions and decimals may be used as entries. The row reduced form supports the rank result. Pivot columns identify independent input directions. The report is useful for algebra homework, engineering models, data transformations, and quick checks before solving systems. Always confirm that rows represent outputs and columns represent inputs. If your course uses another convention, adjust the interpretation before using the result.

Practical Checks

Use small matrices first to see the pattern. Then test larger models after verifying each row. Compare rank with rows and columns. This quickly reveals whether the transformation is one to one, onto, neither, or both. Exported files help save steps for class notes and reports without extra copying.

FAQs

What is the domain of a matrix transformation?

The domain is the input vector space. For an m × n matrix, the domain is Rⁿ because each input vector needs n entries.

What is the codomain of a matrix transformation?

The codomain is the declared output vector space. For an m × n matrix, the codomain is R^m because the product Ax has m entries.

Does rank change the codomain?

No. Rank describes the dimension of the range inside the codomain. The codomain is still determined by the number of matrix rows.

What does nullity mean here?

Nullity is the dimension of the kernel. It counts independent input directions that map to the zero vector under the matrix transformation.

When is the transformation injective?

The transformation is injective when rank equals the number of columns. Then the kernel contains only the zero vector.

When is the transformation surjective?

It is surjective onto the full codomain when rank equals the number of rows. Then every vector in the codomain can be reached.

Can I enter fractions?

Yes. You can enter values like 1/2, -3/4, integers, or decimals. Separate entries with spaces or commas.

Why does a 3 × 2 matrix map R squared to R cubed?

It has two columns, so inputs need two coordinates. It has three rows, so outputs have three coordinates.