2.5 Invertible linear transformations and their properties
3 min read•Last Updated on August 16, 2024
Invertible linear transformations are a key concept in linear algebra. They're special because they have both left and right inverses, preserving important properties of vector spaces like dimension and linear independence.
Understanding invertible transformations helps us grasp how linear maps can be undone. This connects to broader ideas in the chapter about how linear transformations behave and their effects on vector spaces.
Invertible Linear Transformations
Definition and Key Characteristics
Top images from around the web for Definition and Key Characteristics
3.6b. Examples – Inverses of Matrices | Finite Math View original
Is this image relevant?
CS 370: Lab 4: Affine Transformations - Part I, Scaling and Rotation View original
Is this image relevant?
linear algebra - Understanding rotation matrices - Mathematics Stack Exchange View original
Is this image relevant?
3.6b. Examples – Inverses of Matrices | Finite Math View original
Is this image relevant?
CS 370: Lab 4: Affine Transformations - Part I, Scaling and Rotation View original
Is this image relevant?
1 of 3
Top images from around the web for Definition and Key Characteristics
3.6b. Examples – Inverses of Matrices | Finite Math View original
Is this image relevant?
CS 370: Lab 4: Affine Transformations - Part I, Scaling and Rotation View original
Is this image relevant?
linear algebra - Understanding rotation matrices - Mathematics Stack Exchange View original
Is this image relevant?
3.6b. Examples – Inverses of Matrices | Finite Math View original
Is this image relevant?
CS 370: Lab 4: Affine Transformations - Part I, Scaling and Rotation View original
Is this image relevant?
1 of 3
Bijective linear transformation between vector spaces with both left and right inverse
Denoted as T^(-1): W → V for a transformation T: V → W
Satisfies T^(-1) ∘ T = I_V and T ∘ T^(-1) = I_W
Must be both injective (one-to-one) and surjective (onto)
Domain and codomain must have the same dimension
Preserves linear independence, spanning sets, and bases
Examples:
Rotation matrices (rotate vectors in 2D or 3D space)
Scaling transformations with non-zero scale factors
Mathematical Properties
Forms a group under composition (General Linear Group GL(V))
Unique inverse for each invertible transformation
Composition of invertible transformations remains invertible
Inverse of a composition: (S ∘ T)^(-1) = T^(-1) ∘ S^(-1)
Preserves vector space properties:
Dimension
Linear independence
Basis characteristics
Examples:
Identity transformation (trivially invertible)
Matrix transpose for orthogonal matrices
Determining Invertibility
Matrix-Based Criteria
Invertible if and only if its matrix representation [T] is invertible