# Scanned Notes

Scanned PDF.

# Cheatsheet

Thruout this cheatsheet, I use Householder notation. That is, matrixes are denoted with capital English letters $A,B,...$, vectors are denoted with lowercase English letters $a,b,...$, and scalars are denoted with Greek letters $\alpha ,\beta ,...$.

## Vector Space Properties

We say a set $V$ over field $F$ with operations addition $+$ and scalar multiplication $\ast$ is a vector space if and only if the following properties hold for all vectors $u,v,w\in V$ and all scalars $\alpha ,\beta \in F$.

Note: A field is the scalars that make up vectors in $V$. Often $F$ is the real numbers $\mathbb{R}$.

• Addition:
  • Closure: $u+v\in V$.
  • Commutativity: $u+v=v+u$.
  • Associativity: $(u+v)+w=u+(v+w)$.
  • Identity: $\mathrm{\exists }0\in V$ such that $u+0=u$.
  • Additive Inverse: $\mathrm{\exists }-u\in V$ such that $u+(-u)=0$.
• Multiplication:
  • Closure: $\alpha \ast u\in V$.
  • Identity: $\mathrm{\exists }1\in F$ such that $1\ast u=u$.
• Distributivity:
  • Vector Additive Distributivity: $\alpha \ast (u+v)=(\alpha \ast u)+(\alpha \ast v)$.
  • Scalar Additive Distributivity: $(\alpha +\beta )\ast u=(\alpha \ast u)+(\beta \ast u)$.
  • Scalar Multiplicative Distributivity: $(\alpha \beta )\ast u=\alpha \ast (\beta \ast u)$.