PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Algebra → Linear Algebra ↓


Linear Transformations


Linear transformations play a vital role in linear algebra and serve as a fundamental concept that we must understand comprehensively. To understand linear transformations, we will go deep into the definitions, properties, and examples to understand how they work in various mathematical situations.

What is a linear transformation?

A linear transformation is essentially a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. If T is a linear transformation from a vector space V to a vector space W, then for any vectors u and v in V, and any scalar c, the following two properties hold:

T(u + v) = T(u) + T(v)
T(C * U) = C * T(U)

These properties ensure that linear transformations preserve the structure of the vector space. This is important because it means that the transformation does not distort or break relationships between vectors.

Examples of linear transformations

1. Scaling

Scaling is one of the simplest types of linear transformation. For example, scaling a vector by 2 doubles its magnitude but keeps its direction the same. If T(v) = 2v, then:

T(u + v) = 2(u + v) = 2u + 2v = T(u) + T(v)
T(c * u) = 2(c * u) = c * (2u) = c * T(u)

Thus, scaling satisfies both linear transformation properties.

2. Rotation

Rotating a vector in the plane by a certain angle is another example of a linear transformation. Consider rotating a vector v by an angle θ. The new vector T(v) is given by:

T(v) = [cos(θ) -sin(θ); sin(θ) cos(θ)] * v

This transformation preserves vector addition and scalar multiplication.

3. Reflection

Reflection across an axis can serve as another example. Reflecting a vector across the x-axis in 2D is a simple linear transformation. If v = (x, y), then:

t(v) = (x, -y)

Again, this satisfies:

T(u + v) = (x + u, -y - v) = T(u) + T(v)
T(c * u) = (cx, -cy) = c * (x, -y) = c * T(u)

Understanding linear transformations with matrices

One of the powerful aspects of linear transformations is that they can be represented as matrices. If a transformation T is applied to a vector v in Rn, it can often be written as a matrix A multiplying the vector v by itself:

T(v) = av

Here, A is a m×n matrix, where m and n are the dimensions of the output and input vector space, respectively.

Visualizing 2D transformation

Suppose we have a matrix A to perform linear transformations on 2D vectors:

A = [AB;CD]

When this matrix multiplies a vector v = [x; y], it produces a new vector:

AV = [a * x + b * y; c * x + d * y]

Let's see visually what happens when we apply the linear transformation represented by this matrix:

In this visualization, the blue square is the original unit square, and the red square represents its transformation. A linear transformation adjusts conditions, sometimes changing vector lengths and orientations but always keeping them linearly proportional.

Kernel and range of linear transform

When dealing with linear transformations, it is important to understand the kernel and range. These two characteristics provide information about the behavior of the transformation.

Kernels

The kernel of a linear transformation T: V → W is the set of all vectors in V that map T to the zero vector in W. Formally, it is defined as:

Kernel(T) = {v in V | T(v) = 0 in W}

The kernel is a subspace of V, and if it contains only the zero vector, the transformation is called injective (or one-to-one).

Category

The range of T is the set of all possible outputs in W that can be obtained by applying T to any vector in V. Formally:

Range(T) = {T(v) | v in V}

The range is a subspace of W. If the range is the whole space W, the transformation is omniprojective (or onto).

Properties of linear transformations

Linear transformations have several important properties that arise from their definition:

  • The composition of linear transformations is a linear transformation.
  • The inverse of a linear transformation (when it exists) is also a linear transformation.
  • The identity transformation is a linear transformation.

Composition

If T: U → V and S: V → W are linear transformations, then their combination S(T(u)) is also a linear transformation from U to W.

Identification

The identity transformation, which maps each vector onto itself, is a linear transformation, since it satisfies the properties of preserving addition and scalar multiplication.

Applications of linear transformations

Linear transformations are not just abstract mathematical concepts; they have practical applications in various fields, such as computer graphics, engineering, machine learning, etc.

Computer graphics

In computer graphics, transformations such as scaling, rotation, and translation are commonly used to manipulate images and models. Matrices are used to apply these transformations efficiently.

Machine learning

In machine learning, linear transformation techniques such as Principal Component Analysis (PCA) are used to reduce the dimensionality of data, make it more manageable, and uncover hidden structures.

Engineering

In engineering, linear transformations are used in systems and signal processing, providing a means of simplifying complex systems into linear models, making calculations more feasible.

Economics

Economists use linear functions to model the relationships between variables and to forecast economic trends through linear regression techniques.

Conclusion

Linear transformations are embedded in the fabric of linear algebra and many applications that affect our daily lives. From rotations and reflections to transformations in multiple dimensions, linear transformations greatly help us understand and control the manipulation of space and data. Whether used in graphics, signal processing, or data science, recognizing their structure and essence allows us to effectively harness their power.


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Linear Transformations

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