LaTeX for Linear Algebra: Complete Formulas and Notation Guide

Introduction

Mastering linear algebra formulas in LaTeX is essential for mathematical typesetting in academic papers, textbooks, and research documents. Linear algebra is fundamental to many areas of mathematics, science, and engineering, from solving systems of equations to understanding transformations and vector spaces.

This complete guide covers everything you need to know about typesetting linear algebra in LaTeX, from basic vectors and matrices to advanced topics like eigenvalues, eigenvectors, and linear transformations. Whether you're working on academic papers, research documents, or educational materials, mastering linear algebra notation in LaTeX will help you create clear, professional mathematical documents.

Vectors

Vectors are fundamental in linear algebra. Here are the most common ways to represent vectors in LaTeX:

Column Vectors

Column vectors are typically written using the bmatrix or pmatrix environment:

v = v_{1} v_{2} v_{3}

v = a b c

Row Vectors

Row vectors can be written as transposed column vectors or using horizontal notation:

v^{T} = (v_{1} v_{2} v_{3})

Vector Operations

Common vector operations include dot product, cross product, and norm:

u \cdot v = u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3}

u \times v = i u_{1} v_{1} j u_{2} v_{2} k u_{3} v_{3}

∥ v ∥ = v_{1}^{2} + v_{2}^{2} + \dots + v_{n}^{2}

Matrices

Matrices are central to linear algebra. LaTeX provides several environments for different matrix types:

Basic Matrices

Use bmatrix for square brackets, pmatrix for parentheses, and vmatrix for determinants:

A = [a_{11} a_{21} a_{12} a_{22}]

A = (a c b d)

det (A) = a c b d = a d - b c

Large Matrices

For larger matrices, use the same environments with more entries:

A = a_{11} a_{21} ⋮ a_{m 1} a_{12} a_{22} ⋮ a_{m 2} \dots \dots ⋱ \dots a_{1 n} a_{2 n} ⋮ a_{mn}

Special Matrices

Identity matrix, zero matrix, and diagonal matrix:

I_{n} = 10 ⋮ 0 01 ⋮ 0 \dots \dots ⋱ \dots 00 ⋮ 1

0 = 000000000

D = d_{1} 00 0 d_{2} 0 00 d_{3}

Matrix Operations

Essential matrix operations in linear algebra:

Matrix Multiplication

C = A B = [\sum_{k = 1}^{n} a_{ik} b_{kj}]

(A B)_{ij} = k = 1 \sum n a_{ik} b_{kj}

Matrix Transpose

A^{T} = [a_{ji}]

Matrix Inverse

A^{- 1} = \frac{1}{det ( A )} adj (A)

A A^{- 1} = A^{- 1} A = I

Determinant

det (A) = j = 1 \sum n (- 1)^{i + j} a_{ij} det (A_{ij})

det (A B) = det (A) det (B)

Trace

tr (A) = i = 1 \sum n a_{ii}

Rank

rank (A) = dim (col (A)) = dim (row (A))

Systems of Linear Equations

Representing systems of linear equations in matrix form:

Matrix Form

A x = b

a_{11} a_{21} ⋮ a_{m 1} a_{12} a_{22} ⋮ a_{m 2} \dots \dots ⋱ \dots a_{1 n} a_{2 n} ⋮ a_{mn} x_{1} x_{2} ⋮ x_{n} = b_{1} b_{2} ⋮ b_{m}

Augmented Matrix

[A ∣ b] = a_{11} a_{21} ⋮ a_{m 1} a_{12} a_{22} ⋮ a_{m 2} \dots \dots ⋱ \dots a_{1 n} a_{2 n} ⋮ a_{mn} ∣ ∣ ∣ ∣ b_{1} b_{2} ⋮ b_{m}

Solution Methods

Gaussian elimination and Cramer's rule:

x_{i} = \frac{det ( A _{i} )}{det ( A )}

where A_i is the matrix formed by replacing the i-th column of A with \mathbf{b}.

Eigenvalues and Eigenvectors

Fundamental concepts in linear algebra for understanding matrix behavior:

Eigenvalue Equation

A v = λ v

Characteristic Equation

det (A - λ I) = 0

det [a - λ c b d - λ] = (a - λ) (d - λ) - b c = 0

Eigenvalue Decomposition

A = P D P^{- 1}

where P contains the eigenvectors as columns and D is a diagonal matrix of eigenvalues.

A = [v_{1} v_{2} \dots v_{n}] λ_{1} 0 ⋮ 0 0 λ_{2} ⋮ 0 \dots \dots ⋱ \dots 00 ⋮ λ_{n} [v_{1} v_{2} \dots v_{n}]^{- 1}

Vector Spaces

Key concepts in vector space theory:

Linear Combination

v = c_{1} v_{1} + c_{2} v_{2} + \dots + c_{n} v_{n}

Span

span {v_{1}, v_{2}, \dots, v_{n}} = {c_{1} v_{1} + c_{2} v_{2} + \dots + c_{n} v_{n} : c_{i} \in R}

Linear Independence

Vectors are linearly independent if:

c_{1} v_{1} + c_{2} v_{2} + \dots + c_{n} v_{n} = 0 ⟹ c_{1} = c_{2} = \dots = c_{n} = 0

Basis and Dimension

dim (V) = ∣ B ∣

where B is a basis for vector space V.

Linear Transformations

Representing linear transformations as matrices:

Transformation Definition

T : V \to W, T (v) = A v

Properties

T (u + v) = T (u) + T (v)

T (c v) = c T (v)

Kernel and Image

ker (T) = {v \in V : T (v) = 0}

im (T) = {T (v) : v \in V}

Rank-Nullity Theorem

dim (ker (T)) + dim (im (T)) = dim (V)

Inner Products and Norms

Inner product spaces and norms:

Dot Product

⟨ u, v ⟩ = u \cdot v = i = 1 \sum n u_{i} v_{i}

Euclidean Norm

∥ v ∥ = ⟨ v, v ⟩ = i = 1 \sum n v_{i}^{2}

Cauchy-Schwarz Inequality

∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥∥ v ∥

Orthogonality

u ⊥ v ⟺ ⟨ u, v ⟩ = 0

Best Practices

1. Use Appropriate Matrix Environments

Choose the right environment: bmatrix for general matrices, pmatrix for coordinates, vmatrix for determinants.

2. Consistent Notation

Use consistent notation throughout: bold lowercase for vectors (\mathbf{v}), uppercase for matrices (A), Greek letters for eigenvalues (\lambda).

3. Proper Spacing

Use appropriate spacing commands: \, for thin space, \; for medium space, \quad for larger spacing.

4. Subscripts and Superscripts

Use clear subscripts for matrix elements: a_{ij} for the element in row i and column j.

5. Display Mode for Important Formulas

Use display math mode (\[...\] or equation environment) for important formulas that should stand out.

Common Mistakes to Avoid

1. Incorrect Matrix Dimensions

Ensure matrix dimensions match when performing operations. Matrix multiplication AB requires the number of columns of A to equal the number of rows of B.

2. Confusing Vector and Matrix Notation

Use \mathbf{v} or \vec{v} for vectors, not plain v which could be confused with a scalar.

3. Missing Parentheses in Operations

Always use parentheses to clarify order of operations: (AB)^T = B^T A^T, not AB^T.

4. Incorrect Determinant Notation

Use \det(A) or |A| for determinants, not det A.