Definition:Linear Transformation/Vector Space
Jump to navigation
Jump to search
Definition
Let $V, W$ be vector spaces over a field (or, more generally, division ring) $K$.
A mapping $A: V \to W$ is a linear transformation if and only if:
- $\forall v_1, v_2 \in V, \lambda \in K: \map A {\lambda v_1 + v_2} = \lambda \map A {v_1} + \map A {v_2}$
That is, a homomorphism from one vector space to another.
Linear Operator on Vector Space
A linear operator on a vector space is a linear transformation from a vector space into itself.
Also known as
A linear transformation is also known as a linear mapping.
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra
- 2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control ... (previous) ... (next): $1.2$: Linear mappings
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.1$: Continuous and linear maps. Linear transformations
- For a video presentation of the contents of this page, visit the Khan Academy.