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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

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  • 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
  • Invertible Matrix Theorem provides equivalent conditions:
    • Full rank
    • Non-zero determinant
    • Trivial null space
  • Rank-nullity theorem application: invertible if rank(T) = dim(V) = dim(W)
  • Examples:
    • 2x2 matrix with determinant ≠ 0 (invertible)
    • Singular matrix with determinant = 0 (not invertible)

Vector Space Approach

  • Invertible if kernel (null space) contains only the zero vector
  • Test for injectivity and surjectivity separately
  • For finite-dimensional spaces, injectivity implies surjectivity (and vice versa) if dimensions are equal
  • Examples:
    • Linear transformation T(x, y) = (2x + y, x - y) (invertible)
    • Projection onto a subspace (not invertible)

Computing Inverses

Matrix Methods

  • Inverse transformation T^(-1) has matrix representation A^(-1)
  • Compute A^(-1) using:
    • Elementary row operations
    • Adjugate method
  • 2x2 matrix inverse formula: [a b; c d]^(-1) = (1/(ad-bc))[d -b; -c a]
  • Examples:
    • Inverting a 3x3 matrix using row reduction
    • Computing inverse of a 2x2 rotation matrix

Function-Based Approaches

  • For transformations defined by formulas, solve T(T^(-1)(x)) = x for T^(-1)(x)
  • Inverse of a composition: (S ∘ T)^(-1) = T^(-1) ∘ S^(-1)
  • Examples:
    • Finding inverse of T(x) = 3x + 2
    • Inverting a linear transformation defined on a basis

Properties of Invertible Transformations

Algebraic Properties

  • Composition of invertible transformations remains invertible
  • Inverse of a product: (AB)^(-1) = B^(-1)A^(-1)
  • Preserves linear independence and basis properties
  • Forms a group under composition (General Linear Group GL(V))
  • Examples:
    • Proving (AB)^(-1) = B^(-1)A^(-1) for 2x2 matrices
    • Showing preservation of linear independence for a scaling transformation

Geometric Interpretations

  • Preserves dimension of vector spaces
  • Maintains relative positions and orientations of vectors
  • Scales volumes by factor of |det(A)|
  • Examples:
    • Visualizing how rotation matrices preserve angles and distances
    • Demonstrating volume scaling for a 3D linear transformation

Invertibility vs Determinant

Determinant as Invertibility Indicator

  • Linear transformation invertible if and only if determinant is non-zero
  • Determinant of inverse: det(A^(-1)) = 1/det(A)
  • Absolute value of determinant represents volume scaling factor
  • For composition: det(ST) = det(S)det(T)
  • Examples:
    • Calculating determinant to check invertibility of a 3x3 matrix
    • Using determinant to find the inverse of a 2x2 matrix

Geometric Significance

  • Sign change in determinant indicates orientation change in transformed space
  • Determinant magnitude relates to scaling of areas/volumes
  • Zero determinant implies dimension reduction (loss of information)
  • Examples:
    • Visualizing how det(A) < 0 reflects a space inversion
    • Demonstrating area scaling for a 2D linear transformation with |det(A)| = 2
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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