Matrix Basis Domain and Codomain Calculator

Calculator Inputs

Transformation matrix A

Rows define codomain entries. Columns define domain entries.

Domain basis B

Optional. Leave blank for identity. Enter basis vectors as columns.

Codomain basis C

Optional. Leave blank for identity. It must match codomain size.

Input vector coordinates

Optional. Values are read as coordinates in the domain basis.

Decimal precision

Zero tolerance

Example Data Table

Input	Example value	Meaning
Matrix A	1 2 3 4 5 6	Maps R² into R³.
Domain basis B	1 1 0 1	Two independent vectors in the domain.
Codomain basis C	1 0 0 0 1 0 0 0 1	Standard basis for the codomain.
Vector	2 3	Coordinates relative to B.
Expected highlights	Rank 2, nullity 0	Output vector is [11, 27, 43].

Formula Used

For an m by n matrix A, the linear map is T: Rⁿ to R^m.

Domain dimension = n. Codomain dimension = m.

Rank is the number of pivot columns after row reduction.

Nullity = n - rank.

Image basis uses the original pivot columns of A.

Kernel basis comes from the solution set of Ax = 0.

For entered bases, the converted matrix is C^-1AB.

For a vector, x = B[x]_B, y = Ax, and [y]_C = C^-1y.

How to Use This Calculator

Enter the transformation matrix A. Use one matrix row per line.
Enter a domain basis if you want nonstandard input coordinates.
Enter a codomain basis if you want nonstandard output coordinates.
Enter an optional input vector with the same length as the domain dimension.
Set precision and tolerance when decimals or near-zero values appear.
Press Calculate to place the result above the form.
Use CSV or PDF buttons to save the current result.

Understanding the Calculator

A matrix can describe a linear transformation between two vector spaces. The number of columns gives the domain dimension. The number of rows gives the codomain dimension. This calculator reads the transformation matrix, optional domain basis, optional codomain basis, and an optional coordinate vector. It then reports the main structural facts behind the mapping.

Basis, Domain, and Codomain

The domain is the space where input vectors live. For an m by n matrix, the domain is normally R^n. The codomain is the target space. It is normally R^m. A basis gives a coordinate language for a vector space. When a domain basis is supplied, the input coordinates are first converted into the standard coordinate system. When a codomain basis is supplied, the output vector is converted into coordinates relative to that codomain basis.

Rank, Kernel, and Image

Rank measures the dimension of the image. It tells how many independent output directions the matrix can produce. Nullity measures the dimension of the kernel. It tells how many independent input directions collapse to zero. The rank nullity relation connects these values. Rank plus nullity equals the domain dimension. This is a central check for a valid linear transformation.

Coordinate Change Insight

The calculator also builds the matrix of the same transformation between chosen bases. It uses the form C inverse times A times B. Here A is the standard matrix. B stores the domain basis vectors as columns. C stores the codomain basis vectors as columns. This converted matrix maps domain basis coordinates directly into codomain basis coordinates.

Practical Use Cases

This tool helps when studying linear algebra, computer graphics, data transformations, differential systems, or engineering models. It can confirm whether a set of vectors spans the image. It can identify free variables in the kernel. It can also show whether custom bases are valid. This makes the result easier to compare with homework, lecture notes, or manual row reduction work.

Good Input Habits

Enter rows on separate lines. Separate values with spaces or commas. Use square matrices for bases. Keep basis columns independent. Review warnings if a basis is singular. Use the example table to test the layout first. Then compare exported records with your saved work later.

FAQs

What is the domain of a matrix transformation?

The domain is the input space. For an m by n matrix, it is R^n because each input vector needs n entries.

What is the codomain of a matrix transformation?

The codomain is the declared output space. For an m by n matrix, it is R^m because each output vector has m entries.

How are basis vectors entered?

Enter basis vectors as columns of the basis matrix. The domain basis must be n by n. The codomain basis must be m by m.

What happens if I leave a basis blank?

The calculator uses the identity matrix. That means standard coordinates are used for that vector space.

What does rank mean here?

Rank is the number of independent output directions. It also equals the dimension of the image of the transformation.

What does nullity mean here?

Nullity is the number of independent input directions that map to zero. It equals the dimension of the kernel.

Why must a basis matrix be invertible?

A basis must contain independent vectors. An invertible square matrix confirms those vectors define valid coordinates.

Can I download the result?

Yes. Use the CSV button for spreadsheet records. Use the PDF button for a simple printable summary.