57. Linear Transformation
A function between vector spaces is a linear transformation if it preserves vector addition and scalar multiplication.
57.1. The two axioms
For any vectors and scalar :
1. Additivity
2. Homogeneity (scalar multiplication)
The two combine into a single condition: for all scalars and vectors.
Example
defined by is linear.
Check additivity: , :
Check homogeneity: , :
57.2. Consequences
Linearity forces to be very rigid. From the two axioms:
- (apply homogeneity with )
- maps lines through the origin to lines through the origin (or to the origin itself)
- maps the origin to the origin
- maps parallelograms to parallelograms (or degenerate shapes)
57.3. Equivalent definition: matrix–vector product
Every linear transformation can be written as for some matrix — see Matrix Representation. This means linear transformation ⇔ matrix.
57.4. Non-examples
- : not linear (translates the origin off itself)
- : not linear (squaring isn’t homogeneous)
- : not linear (product of components)
57.5. See also
- Matrix–Vector Product — every matrix is a linear transformation
- Matrix Representation — every linear transformation is a matrix
- Image / Kernel — output range / null space
- Composition — composing linear transformations
- Rotation, Reflection, Scaling, Shear