Definition and Basic Properties
Linear Transformation Overview
Definition: A linear transformation T: V → W respects addition and scalar multiplication. Formally: T(u+v)=T(u)+T(v), T(αv)=αT(v) ∀u,v∈V, α∈F.
Invertible Transformation Definition
Definition: T: V → W invertible if ∃ S: W → V such that S∘T = id_V and T∘S = id_W. S is inverse of T, denoted T⁻¹.
Fundamental Properties
Invertibility implies bijection: one-to-one and onto. Linear structure preserved both ways. Dimensions of V and W must be equal for invertibility.
Bijectivity and Invertibility
Injectivity (One-to-One)
Injective: T(v) = T(u) ⇒ v = u. Equivalently, ker(T) = {0}.
Surjectivity (Onto)
Surjective: ∀ w ∈ W, ∃ v ∈ V such that T(v) = w. Image of T equals W.
Equivalence to Invertibility
Theorem: T invertible ⇔ T bijective. Proof relies on existence of inverse mapping preserving linearity.
Matrix Representation
Matrix of a Linear Transformation
Given bases B of V and C of W, T represented by matrix A: W = T(V) ↔ [T]_B^C = A.
Invertibility via Matrices
T invertible ⇔ A invertible (n×n matrix, det(A) ≠ 0). Inverse transformation corresponds to A⁻¹.
Change of Basis Effects
Matrix of inverse changes according to basis transformations: If P and Q are change-of-basis matrices, then new matrix is Q⁻¹AP.
Kernel and Image
Kernel (Null Space)
Definition: ker(T) = {v ∈ V : T(v) = 0}. Measures failure of injectivity. For invertible T, ker(T) = {0}.
Image (Range)
Definition: im(T) = {w ∈ W : w = T(v) for some v ∈ V}. Measures coverage of codomain. For invertible T, im(T) = W.
Rank-Nullity Theorem
dim(V) = dim(ker(T)) + dim(im(T)). For invertible T, rank = dim(V), nullity = 0.
Isomorphisms Between Vector Spaces
Definition of Isomorphism
Isomorphism: invertible linear transformation between vector spaces. Implies vector spaces are structurally identical.
Dimension Criterion
Two finite-dimensional vector spaces over field F are isomorphic if and only if they have equal dimension.
Consequences
Isomorphic spaces share all linear algebraic properties: bases, dimension, linear independence, span, etc.
Existence and Uniqueness of Inverses
Existence Conditions
Inverse exists iff T is bijective. Key proof uses construction of inverse by assigning preimages uniquely.
Uniqueness of Inverse
Inverse is unique: Suppose S and S' both inverses, then S = S'.
Construction Methods
Invert inverse by solving linear systems, matrix inversion, or using rank and kernel computations.
Composition and Invertibility
Composition of Transformations
Given T: U → V and S: V → W, composite S∘T: U → W is linear.
Invertibility of Composition
If T and S invertible, then S∘T invertible with inverse T⁻¹∘S⁻¹.
Associativity and Identity
Composition is associative. Identity transformation id_V is invertible with id_V⁻¹ = id_V.
Properties of Invertible Transformations
Preservation of Structure
Preserve linear combinations, dimension, independence, span, rank, and nullity.
Group Structure
Set of invertible linear transformations on V forms a group under composition: the general linear group GL(V).
Continuity and Topology
Over ℝ or ℂ, invertible linear maps are homeomorphisms in normed vector spaces.
Applications in Linear Algebra
Solving Linear Systems
Invertible transformations correspond to nonsingular coefficient matrices. Unique solution x = A⁻¹b.
Change of Coordinates
Used to switch between bases via invertible matrices, simplifying computations.
Diagonalization and Similarity
Invertible transformations relate similar matrices: A and B are similar if ∃ invertible P with B = P⁻¹AP.
Examples of Invertible Transformations
Identity Transformation
id_V: V → V, id_V(v) = v. Trivially invertible with inverse itself.
Permutation Matrices
Rearranges coordinates. Invertible, inverse is transpose.
Scaling by Nonzero Scalar
T(v) = αv with α ≠ 0, invertible with inverse T⁻¹(v) = (1/α)v.
Rotation in ℝ²
Rotation matrices are invertible orthogonal matrices with inverse equals transpose.
Matrix Example
Matrix A = [[2, 3], [1, 4]] invertible since det(A) = 5 ≠ 0.
Non-Invertible Transformations
Singular Matrices
Matrix A singular if det(A) = 0, no inverse matrix exists.
Zero Transformation
Maps all vectors to zero vector. Kernel = V, image = {0}. Not invertible.
Rank Deficiency
If rank(T) < dim(V), T is not surjective or injective, thus not invertible.
Computational Aspects
Matrix Inversion Algorithms
Gaussian elimination: O(n³) complexity. LU decomposition and adjugate matrix methods also used.
Numerical Stability
Ill-conditioned matrices cause unstable inverses. Condition number measures sensitivity.
Software and Tools
Libraries: LAPACK, MATLAB inv(), NumPy linalg.inv(), symbolic computation in Mathematica.
| Algorithm | Complexity | Notes |
|---|---|---|
| Gaussian Elimination | O(n³) | Standard direct method |
| LU Decomposition | O(n³) | Efficient for multiple solves |
| Adjugate Formula | Expensive for large n | Theoretical, not practical |
Given A ∈ M_n(F), inverse A⁻¹ satisfies:A A⁻¹ = I_nCompute via:1. Augment [A|I_n]2. Row reduce to [I_n|A⁻¹]References
- Axler, S., "Linear Algebra Done Right," Springer, 3rd ed., 2015, pp. 45-78.
- Halmos, P. R., "Finite-Dimensional Vector Spaces," Springer, 2nd ed., 1974, pp. 100-130.
- Lang, S., "Linear Algebra," Springer, 3rd ed., 1987, pp. 60-90.
- Lax, P., "Linear Algebra and Its Applications," Wiley, 2nd ed., 2007, pp. 150-190.
- Strang, G., "Introduction to Linear Algebra," Wellesley-Cambridge Press, 5th ed., 2016, pp. 120-160.