Definition:Linear Transformation
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General Definition
A linear transformation is a homomorphism from one module to another.
Linear Operator
A linear operator is a linear transformation from a module into itself.
Definition in a Vector Space
Let $V, W$ be vector spaces over a field (or, more generally, division ring) $K$.
A mapping $A: V \to W$ is said to be a linear transformation or a linear mapping iff:
- $\forall v_1, v_2 \in V, \lambda \in K: A \left({\lambda v_1 + v_2}\right) = \lambda A \left({v_1}\right) + A \left({v_2}\right)$
Linear Operator
When in fact $V = W$, a linear transformation is called a linear operator.
Some authors, specifically in the field of functional analysis, use the term linear operator (or even just operator) for arbitrary linear transformations.
Sources
- Seth Warner: Modern Algebra (1965): $\S 28$
- For a video presentation of the contents of this page, visit the Khan Academy.
