Vectors in Rⁿ: Definitions & Notations

Vectors in $\mathbb{R}^n$

• Deals with vectors in $\mathbb{R}^n$, defining basic vector operations like addition and scalar multiplication.
• Explores algebraic properties and concepts: dot product, norm, and angle between vectors.

Definitions and Notations

• A vector in $\mathbb{R}^n$ is an ordered list of $n$ real numbers.
  • These numbers $v_1, v_2, \dots, v_n$ are components of the vector.
  • Denoted as $\vec{v}=(v_1, v_2, \dots, v_n)$ where $v_1, v_2, \dots, v_n \in \mathbb{R }$.
• $\mathbb{R}^n$ is the set of all $n$-component real vectors: $\mathbb{R}^n = {(v_1, v_2, \dots, v_n) \mid v_1, v_2, \dots, v_n \in \mathbb{R} }$.
• $\vec{v} = (1, 2, 3)$ is a vector in $\mathbb{R}^3$; $\vec{w} = (-1.5, 0, \sqrt{2}, \pi)$ is a vector in $\mathbb{R}^4$.
• Two vectors $\vec{v}=(v_1, v_2, \dots, v_n)$ and $\vec{w}=(w_1, w_2, \dots, w_n)$ in $\mathbb{R}^n$ are equal if $v_1 = w_1, v_2 = w_2, \dots, v_n = w_n$.
• The zero vector in $\mathbb{R}^n$ is denoted as $\vec{0} = (0, 0, \dots, 0)$.

Vector Operations

• The sum of vectors $\vec{v}=(v_1, v_2, \dots, v_n)$ and $\vec{w}=(w_1, w_2, \dots, w_n)$ in $\mathbb{R}^n$:
  • $\vec{v} + \vec{w} = (v_1 + w_1, v_2 + w_2, \dots, v_n + w_n)$.
• For example, $(1, 2) + (3, 4) = (4, 6)$ and $(1, 2, 3) + (4, 5, 6) = (5, 7, 9)$.
• Scalar multiplication of vector $\vec{v}=(v_1, v_2, \dots, v_n)$ in $\mathbb{R}^n$ by a scalar $c$:
  • $c\vec{v} = (cv_1, cv_2, \dots, cv_n)$
• For example, $2(1, 2) = (2, 4)$ and $-1(1, 2, 3) = (-1, -2, -3)$.