MAT188 Week 1-2: Vectors, Sets, Lines and Planes

Vocabulary flashcards covering core linear algebra definitions for MAT188 Week 1-2.

Vector

A directed line segment starting from a point $P$ and ending at a point $Q$, which represents the displacement from $P$ to $Q$.

Standard Position

The position of a vector when it is represented by an arrow starting from the origin.

Position Vector

The vector $\vec{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$ represented in standard position whose endpoint is at $(x_1, x_2)$.

Euclidean Real Vector Space ($\mathbb{R}^n$)

The collection of all real column vectors with $n$ components, denoted by $\mathbb{R}^n = \left\{ \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} \mid a_1, \dots, a_n \in \mathbb{R} \right\}$.

Real Column Vector

An $n \times 1$ matrix $\begin{pmatrix} v_1 \\ \vdots \\ v_n \end{pmatrix}$, where $v_i \in \mathbb{R}$ for $1 \le i \le n$.

Real Row Vector

A $1 \times n$ matrix $\begin{pmatrix} v_1 & v_2 & \dots & v_n \end{pmatrix}$, where $v_i \in \mathbb{R}$ for $1 \le i \le n$.

Standard Vector ($\vec{e}_i$)

A unit vector in $\mathbb{R}^m$ defined with zeros in all entries except for a $1$ in the $i$-th entry.

Vector Addition

An operation on two vectors $\vec{v}$ and $\vec{w}$ in $\mathbb{R}^n$ defined componentwise as $\vec{v} + \vec{w} = \begin{pmatrix} v_1 + w_1 \\ \vdots \\ v_n + w_n \end{pmatrix}$.

Scalar Multiplication

An operation multiplying a scalar $k \in \mathbb{R}$ and a vector $\vec{v} \in \mathbb{R}^n$ componentwise as $k\vec{v} = \begin{pmatrix} kv_1 \\ \vdots \\ kv_n \end{pmatrix}$.

Parallel Vectors

Two vectors $\vec{v}$ and $\vec{w}$ where one is a scalar multiple of the other (i.e., $\vec{v} = k\vec{w}$ for some $k \in \mathbb{R}$ or $\vec{w} = k\vec{v}$ for some $k \in \mathbb{R}$).

Norm (Magnitude or Length)

For a vector $\vec{v} = \begin{pmatrix} v_1 \\ \vdots \\ v_n \end{pmatrix} \in \mathbb{R}^n$, the quantity denoted by $\|\vec{v}\|$ and defined as $\sqrt{v_1^2 + v_2^2 + \dots + v_n^2}$.

Dot Product

For vectors $\vec{v}$ and $\vec{w}$ with components $v_1, \dots, v_n$ and $w_1, \dots, w_n$, the scalar defined as $\vec{v} \cdot \vec{w} = v_1 w_1 + v_2 w_2 + \dots + v_n w_n$.

Angle Between Vectors

Given vectors $\vec{v}$ and $\vec{w}$, the angle defined as $\arccos\left( \frac{\vec{v} \cdot \vec{w}}{\|\vec{v}\| \|\vec{w}\|} \right)$.

Perpendicular (Orthogonal) Vectors

Two vectors $\vec{v}$ and $\vec{w}$ whose dot product equals zero ($\vec{v} \cdot \vec{w} = 0$).

Set

A collection of objects, called elements or members of the set.

Subset

A set $A$ is a subset of $B$ (written $A \subseteq B$) if every element $a \in A$ is also in $B$.

Equality of Sets

Sets $A$ and $B$ are equal ($A = B$) if $A \subseteq B$ and $B \subseteq A$.

Union of Sets

For subsets $A$ and $B$ of $X$, the set $A \cup B = \{x \in X \mid x \in A \text{ or } x \in B\}$ containing all elements of $A$ and $B$.

Intersection of Sets

For subsets $A$ and $B$ of $X$, the set $A \cap B = \{x \in X \mid x \in A \text{ and } x \in B\}$ containing all common elements between $A$ and $B$.

Direction Vector

A nonzero vector on a line (with tail and tip on the line) that determines the line's direction.

Line in $\mathbb{R}^n$

Any set of the form $L = \{t\vec{m} + \vec{b} \mid t \in \mathbb{R}\}$, where $\vec{b}$ and nonzero $\vec{m}$ are fixed vectors in $\mathbb{R}^n$.

Normal Vector

A nonzero vector $\vec{n}$ that is perpendicular to a given plane.

Plane in $\mathbb{R}^n$

A set of the form $P = \{t\vec{m} + s\vec{n} + \vec{b} \mid s, t \in \mathbb{R}\}$, where nonzero $\vec{m}$, nonzero $\vec{n}$, and $\vec{b}$ are fixed vectors in $\mathbb{R}^n$, and $\vec{m}$ and $\vec{n}$ are not parallel.