Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Algebra → Linear Algebra ↓


Linear Transformations


Linear transformations are a key concept in linear algebra, which itself is a fundamental area of mathematics. Understanding linear transformations involves figuring out how functions map vectors from one vector space to another while preserving the linear structure of the spaces.

What is a linear transformation?

A linear transformation is a function 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 another vector space W, then it satisfies the following properties for all vectors u, v and any scalar c in V:

  • Additivity: T(u + v) = T(u) + T(v)
  • Scalar multiplication: T(c * u) = c * T(u)

Basic examples of linear transformations

Let's start with simple examples. Consider the function T that maps every vector from ℝ² onto itself:

Example 1: Rotation

A common example of a linear transformation is rotation. Suppose we have a 2D vector (x, y), and we want to rotate it by an angle θ. The transformation can be defined by the matrix:

T(x, y) = [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]] * [[x], [y]]
T(x, y) = [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]] * [[x], [y]]

When applied, this transformation rotates the vector counterclockwise by an angle θ around the origin.

Original Rotated (0, 0)

Example 2: Scaling

Scaling is another type of linear transformation. Consider a scaling transformation S that multiplies each vector by a scalar factor. If we apply it to a vector (x, y) in ℝ², we can define it as:

S(x, y) = [[a, 0], [0, b]] * [[x], [y]]
S(x, y) = [[a, 0], [0, b]] * [[x], [y]]

Here a and b are constants, scaling the vector along the x-axis and y-axis, respectively.

Original flaky (0, 0)

Properties of linear transformations

Linear transformations have several important properties that often help simplify problems in linear algebra:

Linearity

The defining property of linear transformations is linearity. They preserve the structure of vector addition and scalar multiplication, which are important for analysis and computation.

Matrix representation

Any linear transformation can be represented by a matrix. If you have a transformation T from ℝ^n to ℝ^m, then there exists a mxn matrix A such that T(v) = A * v for all vectors v in ℝ^n.

For example, if A is a matrix representing a linear transformation, and v is a vector:

A = [[2, 0], [0, 3]] v = [[x], [y]] T(v) = A * v = [[2x], [3y]]
A = [[2, 0], [0, 3]] v = [[x], [y]] T(v) = A * v = [[2x], [3y]]

In this case, the transformation scales the x-component by 2 and the y-component by 3.

Kernel and image

Two important concepts for understanding the nature of linear transformations are the kernel and the image. The kernel of a linear transformation is the set of all vectors that map to the zero vector, and the image is the set of all possible outputs of the transformation.

Reversibility

A linear transformation is invertible if there exists another transformation that can reverse it. If T is from V to W and is invertible, then there exists a transformation T-1 from W to V such that T(T-1 (w)) = w for all w in W and T-1 (T(v)) = v for all v in V

Visualizing linear transformations

Visual examples can help explain how linear transformations work. Consider stretching along an axis:

stretched Original

In this view, the rectangle represents a set of vectors. The transformation stretches the vectors along the x-axis, changing the shape without moving the base point (the origin).

Conclusion

Linear transformations are fundamental in mathematics because they provide a structural and analytical route to understanding vector spaces and matrices. By preserving the vector operations of addition and scalar multiplication, linear transformations allow vast simplifications in calculations. They facilitate the translation of geometric insights and real-world phenomena into mathematical expressions.

The power of linear transformations lies in their ability to represent complex processes in a variety of applications - from computer graphics and engineering to physics and data analysis - unifying diverse fields through mathematical abstraction and representation.


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Linear Transformations

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