Table of Contents
- Introduction
- Scalars, Vectors, and Vector Spaces
- Linear Combinations and Span
- Linear Independence and Basis
- Matrices and Matrix Operations
- Linear Transformations
- The Rank of a Matrix
- Systems of Linear Equations and Gaussian Elimination
- Determinants and Their Properties
- Inverse of a Matrix
- Eigenvalues and Eigenvectors
- Diagonalization and Jordan Form
- Inner Product Spaces and Orthogonality
- Gram-Schmidt Process and Orthonormal Bases
- Applications in Physics and Data Science
- Conclusion
1. Introduction
Linear algebra is the study of vectors, vector spaces, and linear transformations between them. It forms the mathematical foundation for much of physics, engineering, computer science, and data science. This article provides a detailed primer on the core ideas of linear algebra, focusing on intuition, structure, and real-world relevance.
2. Scalars, Vectors, and Vector Spaces
- A scalar is a real (or complex) number.
- A vector is an ordered list of numbers (components), often interpreted as a direction and magnitude.
- A vector space is a set of vectors closed under vector addition and scalar multiplication.
A set \( V \) is a vector space if for any \( \vec{u}, \vec{v} \in V \) and scalar \( c \in \mathbb{R} \), we have:
\[
\vec{u} + \vec{v} \in V, \quad c \vec{v} \in V
\]
3. Linear Combinations and Span
A linear combination of vectors \( \vec{v}_1, \dots, \vec{v}_n \) is:
\[
a_1 \vec{v}_1 + a_2 \vec{v}_2 + \dots + a_n \vec{v}_n
\]
The span of a set is the collection of all linear combinations of those vectors. It forms a subspace of the vector space.
4. Linear Independence and Basis
Vectors are linearly independent if:
\[
a_1 \vec{v}_1 + a_2 \vec{v}_2 + \dots + a_n \vec{v}_n = \vec{0} \Rightarrow a_1 = a_2 = \dots = a_n = 0
\]
A basis is a linearly independent set of vectors that spans the space. The number of basis vectors is the dimension.
5. Matrices and Matrix Operations
A matrix is a rectangular array of numbers that represents a linear transformation. Key operations:
- Addition: element-wise
- Scalar multiplication: scaling every element
- Matrix multiplication: composition of transformations
- Transpose: swapping rows and columns
6. Linear Transformations
A linear transformation \( T: \mathbb{R}^n \to \mathbb{R}^m \) satisfies:
\[
T(a\vec{v} + b\vec{w}) = aT(\vec{v}) + bT(\vec{w})
\]
Every linear transformation can be represented by a matrix.
7. The Rank of a Matrix
The rank of a matrix is the dimension of the image (range) of its associated linear transformation.
- Equals the number of linearly independent rows or columns
- Determines the number of solutions to linear systems
8. Systems of Linear Equations and Gaussian Elimination
A linear system can be written as \( A \vec{x} = \vec{b} \)
- Gaussian elimination transforms \( A \) into row echelon form
- Back substitution finds solutions
The number of solutions depends on the rank and consistency of the system.
9. Determinants and Their Properties
The determinant \( \det(A) \) is a scalar associated with a square matrix.
- \( \det(A) = 0 \): matrix is singular (not invertible)
- \( \det(AB) = \det(A)\det(B) \)
It also gives the scaling factor for volume under transformation.
10. Inverse of a Matrix
A square matrix \( A \) is invertible if there exists a matrix \( A^{-1} \) such that:
\[
AA^{-1} = A^{-1}A = I
\]
Used to solve systems of equations: \( \vec{x} = A^{-1} \vec{b} \)
11. Eigenvalues and Eigenvectors
For a matrix \( A \), an eigenvector \( \vec{v} \) satisfies:
\[
A\vec{v} = \lambda \vec{v}
\]
Where \( \lambda \) is the eigenvalue. They describe scaling directions under transformations.
12. Diagonalization and Jordan Form
If a matrix has \( n \) linearly independent eigenvectors, it can be diagonalized:
\[
A = PDP^{-1}
\]
Where \( D \) is diagonal and \( P \) contains eigenvectors. Otherwise, Jordan canonical form is used.
13. Inner Product Spaces and Orthogonality
An inner product defines angles and lengths:
\[
\langle \vec{u}, \vec{v} \rangle = \sum u_i v_i
\]
Two vectors are orthogonal if their inner product is zero.
14. Gram-Schmidt Process and Orthonormal Bases
Used to convert a linearly independent set into an orthonormal basis:
- Orthogonal: vectors at right angles
- Normalized: unit length
Useful in numerical methods and quantum mechanics.
15. Applications in Physics and Data Science
- Quantum mechanics: state vectors and operators
- Classical mechanics: moment of inertia tensors
- Machine learning: PCA and dimensionality reduction
- Computer graphics: transformations and projections
- Signal processing: Fourier analysis via linear algebra
16. Conclusion
Linear algebra is a foundational tool in both theoretical and applied sciences. Mastery of vectors, matrices, transformations, and eigen-decomposition enables powerful analysis across physics, data science, engineering, and beyond.
