Linear Transform

A function T that maps vectors from Vector Space V into another Vector Space W given by a transformation.

Definition

• Suppose $(V,⊞,⊡)$ and $(W,\oplus ,\odot )$ are vector spaces. A linear transformation function of $T:V\to W$ is a function that follows the axioms:
• Additivity $\forall u,v\in V,T(u⊞v)=T(u)\oplus T(v)$
• Homogenity $\forall v\in V,\forall a\in \mathbb{F},T(a⊡v)=a\odot T(v)$

Implications

This has the further effect of:

• Ensuring all lines are lines (No linear transformation should multiply variables together)
• Ensuring the origin remains at the same position ($T(\overrightarrow{0})=\overrightarrow{0}$ always)

Theorems

• Linear Transformations Preserve Additive Identities
• Linear Transformations Preserve Addition and Scaling
• Linear Transformations Closed Under Linear Combinations
• Linear Maps Determined by Their Actions on A Basis
• Linear Transformation Characterized by Finite Number of Scalars
• Linear Map Lemma
• Linear Map Lemma Independence Corollary

Applications

• Proving A Transformation Is Linear

Concepts

• Matrix Transformation
• Transformation Matrix
• Standard Matrix
• Zero Transformation
• Identity Transformation
• Linear Transformation Rotation
  • Matrix Representation of Linear Rotation
• Linear Map Projection
• Set of Linear Transformations
• Linear Map Composition
• Linear Map Inverse
• Multidimension Linear Transformation