Transformation Matrix Calculator

Linear Transformation:

\[ T: \vec{v} \rightarrow A\vec{v} \]

Transformed Vector (A·v):

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1. What is a Transformation Matrix?

A transformation matrix represents a linear transformation that maps vectors to other vectors in a vector space. The matrix A defines how the vector v is transformed to a new vector Av.

2. How Does the Calculator Work?

The calculator performs matrix-vector multiplication:

\[ A\vec{v} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} a_{11}v_1 + a_{12}v_2 \\ a_{21}v_1 + a_{22}v_2 \end{pmatrix} \]

Where:

\( A \) — 2×2 transformation matrix
\( \vec{v} \) — 2D input vector
\( A\vec{v} \) — Transformed output vector

Explanation: Each component of the output vector is the dot product of a row of the matrix with the input vector.

3. Importance of Linear Transformations

Details: Linear transformations are fundamental in computer graphics, physics simulations, machine learning, and many areas of engineering and mathematics.

4. Using the Calculator

Tips: Enter the 2×2 transformation matrix and a 2D vector. The calculator will compute the transformed vector. Default values represent the identity transformation.

5. Frequently Asked Questions (FAQ)

Q1: What does the identity matrix do?
A: The identity matrix leaves vectors unchanged (A = [[1,0],[0,1]]).

Q2: Can this represent rotations?
A: Yes, rotation by angle θ uses matrix [[cosθ, -sinθ], [sinθ, cosθ]].

Q3: What about scaling transformations?
A: Scaling uses a diagonal matrix [[sx,0],[0,sy]] where sx,sy are scale factors.

Q4: Can I use this for 3D vectors?
A: This calculator handles 2D only. For 3D, you'd need a 3×3 matrix and 3D vector.

Q5: How are these used in computer graphics?
A: Transformation matrices are used to rotate, scale, and translate objects in 2D/3D space.