# Category:Linear Transformations

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This category contains results about **Linear Transformations**.

Definitions specific to this category can be found in Definitions/Linear Transformations.

A **linear transformation** is a homomorphism from one module to another.

Hence, let $R$ be a ring.

Let $M = \struct {G, +_G, \circ}_R$ and $N = \struct {H, +_H, \otimes}_R$ be $R$-modules.

Let $\phi: G \to H$ be a mapping.

Then $\phi$ is a **linear transformation** if and only if:

- $(1): \quad \forall x, y \in G: \map \phi {x +_G y} = \map \phi x +_H \map \phi y$
- $(2): \quad \forall x \in G: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$

## Subcategories

This category has the following 16 subcategories, out of 16 total.

### A

### B

### C

- Compact Linear Transformations (13 P)
- Continuous Transformations (5 P)

### E

### L

- Linear Forms (empty)

### P

- Principle of Superposition (5 P)

## Pages in category "Linear Transformations"

The following 82 pages are in this category, out of 82 total.

### C

- Cesàro Summation Operator is Continuous Linear Transformation
- Change of Coordinate Vectors Under Linear Transformation
- Characterization of Continuous Linear Transformations between Locally Convex Spaces
- Characterization of Norm-to-Weak and Weak-to-Weak Continuity of Linear Transformations
- Characterization of Open Linear Transformation between Normed Vector Spaces
- Complex Conjugation is not Linear Mapping
- Composition of Continuous Linear Transformations is Continuous Linear Transformation
- Composition of Linear Transformations is Isomorphic to Matrix Product
- Composition of Linear Transformations is Linear Transformation
- Condition for Linear Transformation
- Continuity of Linear Transformation/Normed Vector Space
- Continuous Linear Transformations form Subspace of Linear Transformations
- Convergent Sequence of Continuous Real Functions is Integrable Termwise
- Convolution Operator is Continuous Linear Transformation

### D

### E

### I

- Image of Balanced Set under Linear Transformation is Balanced
- Image of Convex Set under Linear Transformation is Convex
- Image of Dilation of Set under Linear Transformation is Dilation of Image
- Image of Generating Set of Vector Space under Linear Transformation is Generating Set of Image
- Image of Linear Transformation is Submodule
- Image of Submodule under Linear Transformation is Submodule
- Image of Translation of Set under Linear Transformation is Translation of Image
- Image of Vector Subspace under Linear Transformation is Vector Subspace
- Infinite Limit Operator is Linear Mapping
- Integrated Linear Differential Mapping is Continuous
- Inverse Evaluation Isomorphism of Annihilator
- Inverse of Linear Transformation is Linear Transformation

### K

- Kernel of Linear Transformation between Finite-Dimensional Normed Vector Spaces is Closed
- Kernel of Linear Transformation contained in Kernel of different Linear Transformation implies Transformations are Proportional
- Kernel of Linear Transformation contains Zero Vector
- Kernel of Linear Transformation is Submodule

### L

- Left Shift Operator is Linear Mapping
- Linear Integral Bounded Operator is Continuous
- Linear Mappings between Vector Spaces form Vector Space
- Linear Transformation as Matrix Product
- Linear Transformation between Normed Vector Spaces is Open iff Image of Open Unit Ball is Open
- Linear Transformation between Topological Vector Spaces Continuous iff Continuous at Origin
- Linear Transformation from Center of Scalar Ring
- Linear Transformation from Finite-Dimensional Vector Space is Injective iff Surjective
- Linear Transformation from Ordered Basis less Kernel
- Linear Transformation is Injective iff Kernel Contains Only Zero
- Linear Transformation is Injective iff Kernel Contains Only Zero/Corollary
- Linear Transformation Maps Zero Vector to Zero Vector
- Linear Transformation of Arithmetic Mean
- Linear Transformation of Generated Module
- Linear Transformation of Submodule
- Linear Transformation of Vector Space Monomorphism
- Linear Transformations between Finite-Dimensional Normed Vector Spaces are Continuous

### P

### R

### S

- Set of Linear Transformations is Isomorphic to Matrix Space
- Set of Linear Transformations over Commutative Ring forms Submodule of Module of All Mappings
- Set of Linear Transformations over Commutative Ring forms Submodule of Module of All Mappings/Unitary
- Set of Linear Transformations under Pointwise Addition forms Abelian Group
- Supremum Operator Norm as Universal Upper Bound
- Supremum Operator Norm is Norm
- Supremum Operator Norm is Well-Defined
- Supremum Operator Norm of Cesàro Summation Operator