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Linear Transformations

Linear-transformation calculator. Enter A and v to compute T(v)=Av with geometric interpretation—for linear transformations.

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T(v) = A v
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4
space-separated

📖 Tutorial｜Linear Transformations

1. Definition
If the matrix of T: Rⁿ→Rᵐ is A in a chosen basis, then T(v)=Av.
2. Symbols
SymbolMeaning
Atransformation matrix
vinput vector
T(v)transformed vector
3. How this tool works
  • Matrix multiplication is a linear transformation;
  • T(αu+βv)=αT(u)+βT(v);
  • Columns of A are coordinates of basis images.
4. Steps
  1. Enter A;
  2. Enter v;
  3. Click Calculate for T(v)=Av.
5. Example
Example: A=1 0;0 −1 (reflection across x-axis), v=(2,3), T(v)=(2,−3).
6. Pitfalls
Columns of A must equal the dimension of v;
Rows of A give the output dimension;
The matrix depends on the chosen basis.

❓ FAQ｜Linear Transformations

What is a linear transformation?
One preserving addition and scaling: T(αu+βv)=αT(u)+βT(v).
Matrix vs transformation?
After choosing a basis, T is represented by A and T(v)=Av.
Rows and columns of A?
Columns = input dimension, rows = output dimension.
Common matrices?
Rotation [[cos,−sin],[sin,cos]], reflection [[1,0],[0,−1]].
What is T(0)?
0; a linear transformation maps zero to zero.