1. Introduction to Vectors and Vector Spaces

 

Introduction to Vectors and Vector Spaces

Vectors in $\mathbb{R}^n$

A vector in $\mathbb{R}^n$ is an ordered list of $n$ real numbers. It can be represented as:

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

where $v_1, v_2, \ldots, v_n$ are the components of the vector $\mathbf{v}$.

Vector Addition and Scalar Multiplication

Vector Addition: If $\mathbf{v} = [v_1, v_2, \ldots, v_n]$ and $\mathbf{w} = [w_1, w_2, \ldots, w_n]$ are vectors in $\mathbb{R}^n$, then their sum $\mathbf{v} + \mathbf{w}$ is given by:

$$ \mathbf{v} + \mathbf{w} = [v_1 + w_1, v_2 + w_2, \ldots, v_n + w_n] $$

Scalar Multiplication: If $\mathbf{v} = [v_1, v_2, \ldots, v_n]$ is a vector in $\mathbb{R}^n$ and $c$ is a scalar, then the scalar multiplication $c\mathbf{v}$ is given by:

$$ c\mathbf{v} = [cv_1, cv_2, \ldots, cv_n] $$

Properties of Vectors

Vectors in $\mathbb{R}^n$ satisfy the following properties:

1. Commutative Law: $\mathbf{v} + \mathbf{w} = \mathbf{w} + \mathbf{v}$ for all vectors $\mathbf{v}$ and $\mathbf{w}$.
2. Associative Law: $(\mathbf{v} + \mathbf{w}) + \mathbf{u} = \mathbf{v} + (\mathbf{w} + \mathbf{u})$ for all vectors $\mathbf{v}$, $\mathbf{w}$, and $\mathbf{u}$.
3. Distributive Law: $c(\mathbf{v} + \mathbf{w}) = c\mathbf{v} + c\mathbf{w}$ for all vectors $\mathbf{v}$ and $\mathbf{w}$, and scalar $c$.
4. Existence of Zero Vector: There exists a vector $\mathbf{0}$ such that $\mathbf{v} + \mathbf{0} = \mathbf{v}$ for all vectors $\mathbf{v}$.
5. Existence of Additive Inverse: For every vector $\mathbf{v}$, there exists a vector $-\mathbf{v}$ such that $\mathbf{v} + (-\mathbf{v}) = \mathbf{0}$.

Vector Spaces

A vector space is a set $V$ together with two operations, vector addition and scalar multiplication, that satisfy the properties of closure, associativity, commutativity, existence of identity and inverses, and distributivity.