Linear Transformation Calculator

Apply a 2×2 linear transformation to a vector and interpret area scaling.

Linear AlgebraTransformationsComputer Science Math

Calculator

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Please enter valid values.

What this calculator teaches

A linear transformation moves vectors while preserving lines and the origin.

The determinant tells how the transformation scales area and whether orientation flips.

Formula & Symbols

ConceptFormulaMeaning
Matrix-vector productT(x,y)=(ax+by,cx+dy)Maps input vector to output vector.
Area scaledet(A)=ad-bcSigned area multiplier.

Step-by-Step Examples

Example 1
A=[[2,1],[0,3]], v=(4,5)
  • x′=2·4+1·5=13.
  • y′=0·4+3·5=15.
✓ T(v)=(13,15).

Where students use this

🎓
College mathematics
Use the tool to connect notation, formulas, and numeric answers.
🧪
Engineering analysis
Apply the result to models, systems, approximations, and design calculations.
💻
Computer science
Use matrix, vector, numerical, and optimization ideas in algorithms and data work.

Common Mistakes to Avoid

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Ignoring restrictions

Check denominator, domain, endpoint, and matrix-invertibility conditions before trusting a result.

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Rounding too early

Carry extra decimals through intermediate steps and round only the final answer.

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Mixing symbolic and numeric answers

Some topics need exact symbolic reasoning, while this calculator gives a clear numerical learning check.

Frequently Asked Questions

Is this calculator meant for students? ▾
Yes. It gives an answer and shows the main method so students can learn the reasoning instead of copying only the result.
Can I use decimals and negative numbers? ▾
Yes, when the topic allows them. Some formulas have restrictions such as positive radii, nonzero denominators, or valid intervals.
Why can textbook answers look different? ▾
Equivalent algebraic forms, different rounding, or using exact notation instead of decimals can make answers look different while meaning the same thing.

Related Math Calculators

🔢 Matrix Calculator🔢 Matrix Diagonalization Calculator🔢 Coordinate Transformation Calculator

Formula Explorer connections

Interpretation: This relationship represents vectors, linear systems or transformations using matrix structure and vector operations. Assumption: Matrix dimensions must be compatible. Inversion, decomposition and unique solutions require rank, conditioning or nonzero-determinant conditions.

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