MATH 110 - Day 2

2024-09-05 12:33:04 -0400 EDT

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Vectors

Definition: Let $ n \in \mathbb{N} = \lbrace x_1, x_2, … x_i | x_i \in \mathbb{R} \text{ for each } i, 1 \leqq i \leqq n \rbrace $

The n-tuples $ ( x:_1, x_2, … x_n) \in \mathbb{R}^n $ are vectors

Examples:

$ \text{for } n = 2, ( x_1, x_2) \in \mathbb{R}^2 $ (2d vector)

$ \text{for } n = 3, ( x_1, x_2, x_3) \in \mathbb{R}^3 $ (3d vector)

Notation: $ \vec{x} = (x_1, x_2, … x_n) \in \mathbb{R}^n $

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Vector Operations

1. Addition

Definition: let $ \vec{v} = (v_1, v_2, … v_n) \in \mathbb{R}^n, \text{ } \vec{w} = (w_1, w_2, … w_n) \in \mathbb{R}^n $

$ \vec{v} + \vec{w} = (v_1 + w_1, v_2 + w_2, … v_n + w_n ) \in \mathbb{R}^n $

Propositions:

1. $ \vec{u} + \vec{v} = \vec{v} + \vec{v} $
2. $ ( \vec{u} + \vec{v} ) + \vec{v} = ( \vec{v} + \vec{v} ) + \vec{v} $
3. $ \vec{u} + \vec{0} = \vec{u} $
4. $ \vec{u} + (-\vec{u}) = 0 $

2. Scaler Multiply

Definition: let $ c \in \mathbb{R} $

$ c\vec{v} = (cv_1, cv_2, … cv_n) \in \mathbb{n} $

Propositions

1. $ c( \vec{u} + \vec{w}) = c\vec{u} + \vec{w} $
2. $ \vec{u}( c + b) = c\vec{u} + b\vec{u} $