Linear Transformation Calculator

Use our free linear transformation calculator to apply a matrix transformation to vectors, see how standard basis vectors transform, batch-transform multiple vectors, and analyze properties like determinant, rank, trace, and invertibility.

Math

Mode

Matrix A (rows; separated)

Vector 𝐯

Multiple vectors (; separated)

Formula

Apply any matrix transformation T(𝐯) = A·𝐯 to vectors. Visualize how standard basis vectors transform, transform multiple vectors at once, and analyze the transformation's properties including determinant, rank, trace, injectivity, surjectivity, and invertibility.

T(𝐯) = A·𝐯, where A is an m×n matrix

Percentage: The percentage value you want to apply

Number: The original number or value

Result: The calculated result

How It Works

1Enter matrix A as rows separated by semicolons, columns by commas (e.g., 1,2;3,4).
2Enter vector(s) for the transformation.
3Select mode: single transform, basis images, multiple vectors, or properties.
4Click Calculate.

Examples

Apply rotation matrix to (1,0)

1.T(1,0) = (0·1+−1·0, 1·1+0·0) = (0,1)

Result: 0

Properties of identity

1.det=1, rank=2, trace=2, invertible ✓

Result: 1

Frequently Asked Questions

A function T(𝐯) = A·𝐯 that maps vectors to vectors while preserving addition and scalar multiplication.

Shows how the standard basis vectors e₁, e₂, etc. transform — these are the columns of A.

det ≠ 0 means the transformation is invertible. det = 0 means it collapses dimension.

The dimension of the image (output space). rank = number of linearly independent columns.

Rows separated by semicolons, columns by commas: e.g., 1,2,3;4,5,6.

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CalculateMe Team

Last updated: 2026-07-15

Frequently Asked Questions

A function T(𝐯) = A·𝐯 that maps vectors to vectors while preserving addition and scalar multiplication.

Shows how the standard basis vectors e₁, e₂, etc. transform — these are the columns of A.

det ≠ 0 means the transformation is invertible. det = 0 means it collapses dimension.

The dimension of the image (output space). rank = number of linearly independent columns.

Rows separated by semicolons, columns by commas: e.g., 1,2,3;4,5,6.

Linear Transformation Calculator

Formula

How It Works

Examples

Apply rotation matrix to (1,0)

Properties of identity

Frequently Asked Questions

What is a linear transformation?

What are basis images?

What does the determinant tell us?

What is the rank?

What format for the matrix?

Related Calculators

Linear Transformation Calculator

Formula

How It Works

Examples

Apply rotation matrix to (1,0)

Properties of identity

Frequently Asked Questions

What is a linear transformation?

What are basis images?

What does the determinant tell us?

What is the rank?

What format for the matrix?

Related Calculators