Operations

Linear algebra operations are the building blocks of data transformations. Whether you’re scaling a vector or multiplying matrices, these operations make abstract ideas concrete—and power everything from graphics to machine learning! ➕➖✖️

Learning Objectives

Learning objectives of the Operations section.

• Matrix Operations
• Vector Operations

Summary Table

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	\(k\mathbf{u} = \begin{bmatrix} ku_1 \\ ku_2 \end{bmatrix}\)