Linear Algebra: Linear transformation
Linear transformation is essentially a “mapping” or “transformation” that moves vectors from one vector space (V) to another vector space (W).
For any transformation T to be considered linear, it must preserve the vector space operations, meaning it must follow two strict rules (known as the linearity conditions):
- Additivity (Sums to Sums): The transformation of a sum of vectors must equal the sum of their individual transformations.
2. Homogeneity (Scalar Products to Scalar Products): The transformation of a vector multiplied by a scalar (a number, c) must equal the scalar multiplied by the vector’s transformation.
These two rules combined also mean that
A linear operator is just a special name used for a linear transformation when the starting vector space and the ending vector space are the same (meaning V=W).
Example 1: Matrix Transformation
Any transformation defined by a matrix
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A is a linear transformation. This is because matrix multiplication inherently follows the two required rules:
Example 2: Specific Matrix Transformations
Specific types of geometric transformations are also linear.
- Reflection about the y-axis.
- Reflection about the x-axis.
- Reflection about the origin.
- Rotation by θ radians about the origin.
Example 3: A Linear Transformation
Example 4: A Nonlinear Transformation
This transformation is nonlinear. Although it satisfies the first property (additivity), it fails the second property (homogeneity) because of the a² term.
Happy learning !!!
