Linear Algebra Tutorial | Generated by AI
1. Introduction
Definition & Importance
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between them. It is fundamental in engineering, physics, computer science, and economics.
Scalars, Vectors, and Matrices
- Scalars: Single numbers (e.g., real or complex numbers)
- Vectors: Ordered lists of numbers, representing magnitude and direction
- Matrices: Rectangular arrays of numbers representing transformations or systems
Applications
- Physics (quantum mechanics, relativity)
- Engineering (control systems, circuits)
- Economics (optimization, game theory)
- Data Science & Machine Learning
2. Systems of Equations
Representation
A system of linear equations can be written in matrix form as: [ Ax = b ] where ( A ) is a matrix, ( x ) is a vector of variables, and ( b ) is a constant vector.
Solution Methods
- Gaussian Elimination: Converts the system to row echelon form to solve for unknowns.
- Row Reduction (Reduced Row Echelon Form, RREF): Further reduces the matrix to identify solutions.
- Solution Types:
- Unique solution: One intersection point
- Infinite solutions: Multiple intersections
- No solution: Parallel lines (inconsistent system)
- Homogeneous vs. Non-Homogeneous:
- Homogeneous: ( Ax = 0 ) (always has at least one solution)
- Non-homogeneous: ( Ax = b )
3. Matrices and Operations
Notation & Types
- Square Matrix: Same number of rows and columns
- Identity Matrix (I): Diagonal elements are 1, others are 0
- Zero Matrix (0): All elements are zero
Operations
- Addition & Subtraction: Element-wise
- Scalar Multiplication: Multiply each element by a scalar
- Matrix Multiplication: ( (AB){ij} = \sum{k} A_{ik} B_{kj} )
- Transpose: Flipping rows and columns
- Inverse (A(^-1)): Exists only if determinant is nonzero
4. Determinants
Definition
A scalar value associated with a square matrix, useful in solving linear equations and understanding matrix properties.
Computation
- 2×2 Matrix: ( \text{det} \begin{bmatrix} a & b \ c & d \end{bmatrix} = ad - bc )
- 3×3 Matrix: Use cofactor expansion or Sarrus’ Rule
- Higher-Order Matrices: Use row expansion or Laplace expansion
Properties & Applications
- Cramer’s Rule: Uses determinants to solve systems ( Ax = b )
- Singular vs. Nonsingular Matrices: Determinant ( = 0 ) means non-invertible
5. Vector Spaces
Definition
A set of vectors that can be added together and multiplied by scalars while remaining within the set.
Key Concepts
- Subspaces: A subset of a vector space satisfying closure properties
- Basis: A minimal set of linearly independent vectors spanning a space
- Dimension: The number of basis vectors
- Linear Independence: A set of vectors where no vector is a linear combination of others
- Span: All possible linear combinations of a given set of vectors
- Change of Basis: Transition between different vector space representations
6. Linear Transformations
Definition
A function ( T: V \to W ) that preserves vector addition and scalar multiplication.
Representation
Every linear transformation can be represented as a matrix.
Properties
- Kernel (Null Space): Set of vectors mapped to zero
- Image (Range): Set of output vectors
- Injectivity (One-to-One): ( \text{Ker}(T) = {0} )
- Surjectivity (Onto): Image spans the entire codomain
7. Eigenvalues and Eigenvectors
Definitions
- Eigenvalues (λ): Scalars where ( Av = \lambda v )
- Eigenvectors (v): Nonzero vectors satisfying ( Av = \lambda v )
Computation
- Solve the characteristic equation ( \det(A - \lambda I) = 0 )
- Find corresponding eigenvectors from ( (A - \lambda I)v = 0 )
Applications
- Stability analysis
- Markov processes
- Principal Component Analysis (PCA) in machine learning
8. Inner Product Spaces
Dot Product & Norms
- Dot Product: ( a \cdot b = \sum a_i b_i )
- Norm (Magnitude): ( |v| = \sqrt{v \cdot v} )
Orthogonality & Orthonormal Bases
- Orthogonal Vectors: Dot product = 0
- Orthonormal Basis: Basis where all vectors are orthogonal and have unit length
- Gram-Schmidt Process: Converts a basis into an orthonormal one
9. Applications
Real-World Problems
- Network Flows: Optimization using linear algebra
- Data Fitting: Regression techniques using least squares
Computational Methods
- Matrices in Programming: Used in Python, MATLAB, and NumPy for scientific computing