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. (Lay, Lay, and McDonald 2015) 📊

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 Linear Algebra section.

Systems of Linear Equations

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	\left[\begin{array}{ccc|c} 2 & -1 & 1 & 8\\ 1 & 0 & 0 & 2\\ 0 & 0 & 4 & 7\end{array}\right]
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}

Operations

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}

Properties

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

Lay, David C., Steven R. Lay, and Judi J. McDonald. 2015. Linear Algebra and Its Applications. 5th ed. Pearson.