Vectors and Vector Spaces Part 1: Foundations and Vector Operations

Vocabulary flashcards covering core definitions and terms from Module 4: Vectors and Vector Spaces Part 1 (Foundations and Vector Operations).

Scalar

A number used to scale a vector, belonging to $\mathbb{R}$ in real vector spaces.

Coordinate Vector

An ordered list of $n$ real numbers in $\mathbb{R}^n$, usually written as a column vector.

Components

The individual entries that make up a coordinate vector, where their order matters.

Coordinate Space ($\mathbb{R}^n$)

The set of all ordered $n$-tuples of real numbers written as column vectors $\begin{pmatrix} x_1 \\ x_2 \\ \dots \\ x_n \end{pmatrix}$ where $x_i \in \mathbb{R}$ for $i = 1, \dots, n$.

Equality of Coordinate Vectors

The condition where two coordinate vectors have the same number of components and all corresponding components are equal.

Displacement Vector

A vector that describes a change in location between points, as opposed to a point which identifies a fixed location.

Vector Addition

An operation on vectors $u, v \in \mathbb{R}^n$ performed by adding corresponding components to yield $u + v = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ \dots \\ u_n + v_n \end{pmatrix}$.

Zero Vector

A vector in $\mathbb{R}^n$ consisting of $n$ zero components, denoted $0 = (0, \dots, 0)^T$, which represents no displacement and satisfies $u + 0 = u$.

Additive Inverse

For a vector $u$, the vector $-u = (-u_1, \dots, -u_n)^T$ that reverses every component, satisfying $u + (-u) = 0$.

Vector Subtraction

The operation defined as the addition of the additive inverse, given by $u - v = u + (-v)$.

Scalar Multiplication

An operation multiplying every component of a vector $u \in \mathbb{R}^n$ by a real scalar $a \in \mathbb{R}$, resulting in $au = \begin{pmatrix} au_1 \\ au_2 \\ \dots \\ au_n \end{pmatrix}$.

Euclidean Norm

The length of a vector $u \in \mathbb{R}^n$, defined as $\|u\| = \sqrt{u_1^2 + u_2^2 + \dots + u_n^2}$.

Euclidean Distance

The distance between two vectors $u$ and $v$, defined as $d(u, v) = \|u - v\| = \sqrt{\sum_{i=1}^n (u_i - v_i)^2}$.

Unit Vector

A vector whose Euclidean norm is equal to $1$.

Normalization

The process of dividing a non-zero vector $u$ by its norm to create a unit vector $\hat{u} = \frac{u}{\|u\|}$ pointing in the same direction.

Real Vector Space

A non-empty set $V$ equipped with vector addition and scalar multiplication by real numbers that satisfies the ten vector-space axioms.

Subspace

A subset $W$ of a vector space $V$ that is itself a vector space using the same addition and scalar multiplication operations as $V$.

Subspace Test

A three-condition criteria proving $W \subseteq V$ is a subspace by showing $0 \in W$, closure under addition ($u + v \in W$), and closure under scalar multiplication ($au \in W$).

Kernel of a Matrix

For an $m \times n$ matrix $A$, the set $\text{ker}(A) = \{x \in \mathbb{R}^n : Ax = 0\}$, which is always a subspace of $\mathbb{R}^n$.

Feature Vector

An ordered vector of numerical measurements $(x_1, \dots, x_n)^T$ representing an object's features in computing applications.