1: Vector definition and operations

Section 1: Vector Definition and Operations

Vectors are fundamental objects in linear algebra, representing quantities possessing both magnitude and direction. In an algebraic context, a vector in $\mathbb{R}^n$ ($n$-dimensional real space) is defined as an ordered list of $n$ real numbers, called its components or coordinates. We typically denote a vector as a column:

$$\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$$

or equivalently as a row: $\mathbf{v} = [v_1, v_2, \dots, v_n]^T$. The integer $n$ is the dimension of the vector. Geometrically in 2D ($\mathbb{R}^2$) or 3D ($\mathbb{R}^3$), a vector is visualized as an arrow from the origin $(0,0)$ or $(0,0,0)$ to the point $(v_1, v_2)$ or $(v_1, v_2, v_3)$. This arrow representation captures direction (the orientation) and magnitude (the length).

Core Vector Operations

1. Vector Addition:
   Given vectors $\mathbf{u} = [u_1, u_2, \dots, u_n]^T$ and $\mathbf{v} = [v_1, v_2, \dots, v_n]^T$ of the same dimension, their sum $\mathbf{u} + \mathbf{v}$ is computed component-wise:

   $$\mathbf{u} + \mathbf{v} = \begin{bmatrix} u_1 + v_1 \\ u_2 + v_2 \\ \vdots \\ u_n + v_n \end{bmatrix}$$

   Geometrically, this corresponds to placing the tail of $\mathbf{v}$ at the head of $\mathbf{u}$ and drawing the arrow from the tail of $\mathbf{u}$ to the head of $\mathbf{v}$ (the parallelogram rule).

2. Scalar Multiplication:
   Given a vector $\mathbf{v} = [v_1, v_2, \dots, v_n]^T$ and a real number $c$ (a scalar), the product $c\mathbf{v}$ is:

   $$c\mathbf{v} = \begin{bmatrix} c v_1 \\ c v_2 \\ \vdots \\ c v_n \end{bmatrix}$$

   Geometrically, this scales the vector's length by $|c|$ and reverses its direction if $c < 0$.

3. Vector Subtraction:
   Subtraction is defined using addition and scalar multiplication: $\mathbf{u} - \mathbf{v} = \mathbf{u} + (-1)\mathbf{v}$. Component-wise:

   $$\mathbf{u} - \mathbf{v} = \begin{bmatrix} u_1 - v_1 \\ u_2 - v_2 \\ \vdots \\ u_n - v_n \end{bmatrix}$$

   Geometrically, $\mathbf{u} - \mathbf{v}$ points from the head of $\mathbf{v}$ to the head of $\mathbf{u}$ when both tails are placed at the origin.

Essential Properties

These operations satisfy key algebraic properties for any vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$ in $\mathbb{R}^n$ and scalars $c, d$:

• Commutativity of Addition: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$
• Associativity of Addition: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$
• Additive Identity: The zero vector $\mathbf{0} = [0, 0, \dots, 0]^T$ satisfies $\mathbf{v} + \mathbf{0} = \mathbf{v}$
• Additive Inverse: For every $\mathbf{v}$, there exists $-\mathbf{v} = (-1)\mathbf{v}$ such that $\mathbf{v} + (-\mathbf{v}) = \mathbf{0}$
• Distributive Laws:
  $c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$
  $(c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v}$
• Scalar Multiplication Associativity: $c(d\mathbf{v}) = (cd)\mathbf{v}$
• Scalar Identity: $1 \cdot \mathbf{v} = \mathbf{v}$