Definition and Basic Concept
Linear Transformations
Definition: A linear transformation T: V → W between vector spaces preserves addition and scalar multiplication. Notation: T(αu + βv) = αT(u) + βT(v).
Matrix Representation
Concept: Represent T as a matrix A relative to chosen bases of V and W, enabling computational manipulation. Purpose: Simplify abstract linear maps into concrete algebraic objects.
Role of Bases
Necessity: Matrix form depends on bases selection. Without bases, T is an abstract operator; with bases, T corresponds to a unique matrix.
Bases and Coordinate Vectors
Definition of Bases
Definition: A basis B for vector space V is a linearly independent spanning set. Dimension: Number of vectors in B equals dim(V).
Coordinate Vectors
Representation: Every vector v ∈ V expressed uniquely as linear combination of basis vectors. Coordinates form a column vector.
Notation
Notation: If B = {v₁, ..., vₙ}, then [v]_B = (c₁, ..., cₙ)ᵀ where v = Σ cᵢ vᵢ.
Construction of the Matrix Representation
Mapping Basis Vectors
Procedure: Apply T to each basis vector of V, express image in W basis. Columns of matrix correspond to these coordinate vectors.
Matrix Columns
Each column: Coordinates of T(vᵢ) in W basis. Matrix size: m×n if dim(W)=m and dim(V)=n.
Formula
A = [ [T(v₁)]_W | [T(v₂)]_W | ... | [T(vₙ)]_W ]Change of Basis and Similarity
Change-of-Basis Matrices
Definition: Matrices P, Q convert coordinates between bases. P converts from old to new basis; Q inverse.
Similarity Transformations
Relation: Matrix of T under new basis equals P⁻¹AP. Similar matrices represent the same linear operator under different bases.
Invariance
Invariants: Eigenvalues, determinant, trace remain constant under similarity. Basis-dependent: matrix entries.
Properties of Matrix Representations
Linearity
Property: Matrix representation respects addition and scalar multiplication of linear maps.
Rank and Nullity
Rank: Rank of matrix equals dimension of image of T. Nullity: Dimension of kernel corresponds to nullity of matrix.
Determinant and Trace
Determinant: Nonzero determinant implies invertibility. Trace: Sum of diagonal elements, related to eigenvalues.
| Property | Matrix Interpretation |
|---|---|
| Invertibility | Matrix is invertible ⇔ linear map is bijective |
| Rank | Rank equals dimension of image space |
| Trace | Sum of eigenvalues, invariant under basis change |
Composition of Linear Transformations
Matrix Multiplication
Rule: If S: U → V and T: V → W, then matrix of T∘S is product of matrices of T and S.
Order of Multiplication
Note: Matrix of T∘S = matrix of T × matrix of S. Matrix multiplication is associative, not commutative.
Formula
[T∘S] = [T] × [S]Invertibility and Matrix Representation
Invertible Linear Maps
Condition: T invertible iff matrix representation A is invertible (nonzero determinant).
Inverse Matrix
Matrix of T⁻¹ is A⁻¹ relative to corresponding bases.
Consequences
Invertibility implies isomorphism between vector spaces. Dimensions of domain and codomain must agree.
Examples of Matrix Representations
Identity Transformation
Matrix: Identity matrix Iₙ. Basis: Any basis yields Iₙ as matrix representation.
Zero Transformation
Matrix: Zero matrix of appropriate size. All vectors map to zero vector.
Rotation in ℝ²
Matrix: [[cos θ, -sin θ], [sin θ, cos θ]] relative to standard basis.
Applications in Linear Algebra
Eigenvalue Computation
Reduction: Matrix representation allows characteristic polynomial derivation and eigenvalue computation.
Diagonalization
Purpose: Find basis where matrix is diagonal, simplifying powers and functions of T.
Solving Systems
Linear systems: Represented as matrix equations Ax = b, solvable by matrix methods.
Computational Aspects
Storage
Matrix form enables storage in arrays, efficient manipulation by algorithms.
Algorithms
Operations: Gaussian elimination, LU decomposition, eigenvalue algorithms rely on matrix form.
Software
Tools: MATLAB, NumPy, Mathematica implement matrix operations for linear transformations.
| Algorithm | Purpose | Complexity |
|---|---|---|
| Gaussian Elimination | Solve Ax = b | O(n³) |
| LU Decomposition | Matrix factorization | O(n³) |
| QR Algorithm | Eigenvalue approximation | O(n³) |
Limitations and Challenges
Basis Dependence
Issue: Matrix representation varies with choice of bases; lacks intrinsic uniqueness.
Computational Complexity
Large dimensions: Matrix sizes grow quadratically; algorithms become costly.
Numerical Stability
Rounding errors: Floating point approximations impact accuracy of matrix computations.
References
- Axler, S., Linear Algebra Done Right, Springer, 3rd Edition, 2015, pp. 34-78.
- Strang, G., Introduction to Linear Algebra, Wellesley-Cambridge Press, 5th Edition, 2016, pp. 120-165.
- Hoffman, K., Kunze, R., Linear Algebra, Prentice-Hall, 2nd Edition, 1971, pp. 50-110.
- Lang, S., Linear Algebra, Springer, 3rd Edition, 1987, pp. 95-140.
- Axler, S., Eigenvalues, Eigenvectors, and Matrix Representations, American Mathematical Monthly, Vol. 95, No. 2, 1988, pp. 117-134.