Section 1 Definitions

Vector
An Object that has the following properties:

  1. $\vec{a}+\vec{b} = \vec{c}$ where $\vec{c}$ is a vector
  2. $c\vec{a} = \vec{d}$ where $\vec{d}$ is a vector
  3. $\vec{a}+\vec{b} = \vec{b}+\vec{a}$ Commutative Property
  4. $\vec{a} + (\vec{b}+\vec{c}) = (\vec{a}+\vec{b}) + \vec{c}$ Associative Property
  5. $\vec{a} * 0 = \vec{0}$
  6. $\vec{a} * 1 = \vec{a}$ Multiplicative Identity
  7. $\vec{a} + (-\vec{a}) = \vec{0}$ Additive Inverse

Vectorspace

A Vectorspace is a non-empty set V of objects with operations addition and scalar multiplication such that:

  1. $\forall$ $\vec{u},\vec{v} \in V, \vec{u}+\vec{v} \in V$ closed
  2. $\forall$ $\vec{u},\vec{v} \in V, \vec{u}+\vec{v} = \vec{v}+\vec{u}$ Commutative Property
  3. $\forall$ $\vec{u},\vec{v}, \vec{w} \in V, \vec{u}+(\vec{v} + \vec{w}) = (\vec{u}+\vec{v})+\vec{w}$ Associative Property
  4. $\exists$ $\vec{0} \in V$ such that $\vec{u}+\vec{0} = \vec{u}$ $\forall$ $\vec{u} \in V$ Additive Identity
  5. $\forall$ $\vec{u} \in V$ $\exists$ $-\vec{u}$ such that $\vec{u}+-\vec{u} = \vec{0}$ Additive Inverse
  6. $\forall$ $\vec{u},\vec{v} \in V$ & $c \in \Re, c(\vec{u})\in V$ Scalar Multiplication
  7. $\forall$ $\vec{u} \in V$ and $c \in \Re , c(\vec{u}+\vec{v}) = c\vec{u}+c\vec{v}$ Distributive Property for Scalars
  8. $\forall$ $c,d \in \Re$ and $\vec{u} \in V, (c+d)\vec{u} = c\vec{u}+d\vec{u}$ Distributive Property for Vectors
  9. $\forall$ $c,d \in \Re$ and $\vec{u} \in V, c(d\vec{u}) = (cd)\vec{u}$ Associative Property of Multiplication
  10. $\forall$ $\vec{u} \in V, 1*\vec{u} = \vec{u}$ Multiplicative Identity

When we think of Vector Spaces, your home is $\Re^n$ for $n\geq 0$, the set of all polynomials $P_n$ of degree less than or equal to n where

(1)
\begin{equation} p(t) = a_0+a_1t+a_2t^2+ ...+a_nt^n \end{equation}

Subspaces
A subspace of a vectorspace V is a subset H of V such that

  1. $\vec{0} \in H$ _ The zero vector must be included
  2. $\forall$ $\vec{u},\vec{v} \in H, \vec{u}+\vec{v} \in H$ _ Any linear combination of vectors defining H must be included in span{H}
  3. $\forall$ $\vec{u} \in H$ and $c \in \Re, c\vec{u} \in H$ ___ Any scalar multiple of a vector in H must be included in H
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