A Linear Transformation Calculator is a computational tool used to apply linear transformations to vectors or matrices. Linear transformations are mathematical operations that map vectors from one vector space to another while preserving vector addition and scalar multiplication. These transformations are often represented by matrices. This tool typically allows users to input a matrix representing the linear transformation and a vector or set of vectors to be transformed, and then it calculates the resulting transformed vectors.
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Formula of a Linear Transformation
The formula for a linear transformation T of a vector v using a matrix A is given by:
\( T(v) = Av \)
Where:
- A is the transformation matrix.
- v is the input vector.
- T(v) is the transformed vector.
For example, if A is a 2x2 matrix and v is a 2-dimensional vector, the transformation is calculated as follows:
\( T \left( \begin{pmatrix} x \\ y \end{pmatrix} \right) = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} a_{11}x + a_{12}y \\ a_{21}x + a_{22}y \end{pmatrix} \)
Where to Use a Linear Transformation Calculator
A Linear Transformation Calculator is useful in various fields, including:
- Mathematics: For solving linear algebra problems, including transformations, eigenvalues, and eigenvectors.
- Physics: For transforming coordinates, especially in mechanics and quantum physics.
- Engineering: For analyzing systems and structures, such as stress transformations and electrical circuits.
How to use linear transformation calculator
- Enter the elements of the transformation matrix in the fields under “Matrix A”:
- For example:
- a11: 2
- a12: 1
- a21: 0
- a22: 1
- For example:
- Enter the components of the vector you want to transform in the fields under “Vector v”:
- For example:
- v1: 3
- v2: 4
- For example:
- Click the “Calculate” button.
- The calculator will compute the transformed vector and display the result on the screen.