Linear Algebra

The language of data, geometry, and computation—linear algebra provides the building blocks for modeling, transforming, and understanding high-dimensional spaces. From systems to structures, it’s at the heart of modern applied math.[@LinearAlgebra] 📊

Learning Objectives

Learning objectives of the Linear Algebra section.

• Systems of Linear Equations
  • Lines
  • Linear Systems
  • Matrices
  • Vectors

• Operations
  • Dot Product
  • Vector Operations

• Properties
  • Norm
  • Transpose
  • Inverse
  • Eigenvalues

Summary Table

Summary of the Systems of Linear Equations section.

Concept	Description	Example
Linear Equation	An equation in the form \(a_1x_1 + a_2x_2 + \cdots + a_nx_n = b\)	\(2x_1 - x_2 + x_3 = 8\)
System of Equations	A set of linear equations with shared variables	\(\begin{aligned} a_1x_1 + b_1x_2 &= c_1 \\ a_2x_1 + b_2x_2 &= c_2 \end{aligned}\)
Coefficient Matrix	A matrix containing only the coefficients of a system’s variables	\(\begin{bmatrix} 2 & -1 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 4 \end{bmatrix}\)
Augmented Matrix	A matrix combining coefficients and constants from a system	\(\begin{bmatrix} 2 & -1 & 1 & 8 \\ 1 & 0 & 0 & 2 \\ 0 & 0 & 4 & 7 \end{bmatrix}\)
Matrix Notation	General form to represent a matrix \(A\) with \(m\) rows and \(n\) columns	\(A = \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{bmatrix}\)
Vector	An ordered list of numbers representing a point or direction in space	\(\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}\)
Column Vector	A vertical vector derived from a matrix column	\(\mathbf{a}_1 = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}\)
Row Vector	A horizontal vector derived from a matrix row	\(\mathbf{a}_1^T = \begin{bmatrix} 2 & -1 & 1 \end{bmatrix}\)

Summary of the Operations section.

Concept	Description	Example
Linear Transformation	A function that maps \(\mathbb{R}^n \rightarrow \mathbb{R}^m\) using a matrix	\(T(\mathbf{x}) = A\mathbf{x}\) where \(A\) is an \(m \times n\) matrix
Matrix-Vector Product	Applies the transformation \(T(\mathbf{x})\) via matrix multiplication	\(A\mathbf{x} = \mathbf{b}\)
Vector Addition	Adds corresponding components (head-to-tail rule)	\(\mathbf{u} + \mathbf{v} = \begin{bmatrix} u_1 + v_1 \\ u_2 + v_2 \end{bmatrix}\)
Vector Subtraction	Subtracts corresponding components (tip-to-tip vector)	\(\mathbf{u} - \mathbf{v} = \begin{bmatrix} u_1 - v_1 \\ u_2 - v_2 \end{bmatrix}\)
Scalar Multiplication	Scales the vector’s length and may reverse its direction	\(k\mathbf{u} = \begin{bmatrix} ku_1 \\ ku_2 \end{bmatrix}\)

Summary of the Properties section.

Concept	Description	Example
L1-norm	Measures total absolute difference between two vectors (Manhattan distance).	\[\| \mathbf{u} - \mathbf{v} \|_1 = \sum_{i=1}^n |u_i - v_i|\]
L2-norm	Measures straight-line distance between two vectors (Euclidean distance).	\[\| \mathbf{u} - \mathbf{v} \|_2 = \sqrt{ \sum_{i=1}^n (u_i - v_i)^2 }\]
Transpose	Flips a matrix over its diagonal, turning rows into columns.	\[A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}, \quad A^\top = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}\]
Inverse (2×2 Matrix)	Reverses a matrix transformation when the determinant is nonzero.	\[A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}, \quad \det(A) = ad - bc\]
Eigenvalues/Vectors	\(\mathbf{v}\) is an eigenvector of \(A\) if \(A \mathbf{v} = \lambda \mathbf{v}\).	\[A \mathbf{v} = \lambda \mathbf{v}\]
Characteristic Equation	Equation whose solutions are the eigenvalues of \(A\).	\[\det(A - \lambda I) = 0\]