Linear Transformation Calculator

This calculator helps you perform linear transformations in 2D space, essential for solving linear algebra problems. It is designed for students, educators, and professionals who need a reliable tool for these calculations.

Calculator

Please enter a valid number for matrix A.
Please enter a valid number for matrix B.
Please enter a valid number for matrix C.
Please enter a valid number for matrix D.
Please enter a valid number for vector X.
Please enter a valid number for vector Y.

Results

Transformed X 0.00
Transformed Y 0.00

Data Source and Methodology

All calculations are based on mathematical principles of linear algebra. For further reading, refer to "Linear Algebra and Its Applications" by Gilbert Strang.

The Formula Explained

\[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \]

Glossary of Variables

  • Matrix A, B, C, D: Coefficients of the transformation matrix.
  • Vector X, Y: Components of the vector being transformed.
  • Transformed X, Y: Resulting components after transformation.

How It Works: A Step-by-Step Example

Consider a transformation matrix \(\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and a vector \(\begin{pmatrix} 5 \\ 6 \end{pmatrix}\). The transformed vector is computed as follows:

\[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 \\ 6 \end{pmatrix} = \begin{pmatrix} 1*5 + 2*6 \\ 3*5 + 4*6 \end{pmatrix} = \begin{pmatrix} 17 \\ 39 \end{pmatrix} \]

Frequently Asked Questions (FAQ)

What is a linear transformation?

A linear transformation is a function that maps a vector space to another vector space in a linear manner.

How do I use this calculator?

Enter the coefficients of your 2x2 matrix and the components of your vector to compute the linear transformation.

Can this calculator handle 3D transformations?

Currently, this calculator is designed for 2D transformations. For 3D, consider using computational software like MATLAB or Python with NumPy.

What if my matrix is not invertible?

If your matrix is singular, it indicates a transformation that does not have a unique inverse.

Why does the calculator display NaN?

Ensure that all inputs are valid numbers. NaN (Not a Number) appears when inputs are invalid or undefined.

Tool developed by Ugo Candido. Content reviewed by the CalcDomain Expert Team.
Last reviewed for accuracy on: October 15, 2023.

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