Vector Spaces

• a vector space over a field $F$ is a set of of vectors where
  • addition is ^cbddf2
  • multiplication is ^85a8b3
• and satisfies the below axioms

Addition §

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Commutativity”,“label”:“commutativity”,“_index”:0}] Axiom 1 (Commutativity). $\vec{u}+\vec{v}=\vec{v}+\vec{u}$.

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Associativty”,“label”:“associativty”,“_index”:1}] Axiom 2 (Associativty). $(\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})$.

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Zero Vector”,“label”:“zero-vector”,“_index”:2}] Axiom 3 (Zero Vector). Exists a vector $\vec{0}$ s.t. $\vec{v}+\vec{0}=\vec{v}$.

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Additive Inverse”,“label”:“additive-inverse”,“_index”:3}] Axiom 4 (Additive Inverse). $\forall \vec{v}\in V$, $\exists \vec{w}$ s.t. $\vec{v}+\vec{w}=\vec{0}$. $\vec{w}=-\vec{v}$.

Multiplication §

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Multiplicative Identity”,“label”:“multiplicative-identity”,“_index”:4}] Axiom 5 (Multiplicative Identity). $\exists 1\in V$ s.t. $1\vec{u}=\vec{u}$.

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Multiplicative Associativity”,“label”:“multiplicative-associativity”,“_index”:5}] Axiom 6 (Multiplicative Associativity). For scalars $c,d$, $c(d\vec{u})=(cd)\vec{u}$.

Distribution §

• connects addition and multiplication

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“First Distribution”,“label”:“first-distribution”,“_index”:6}] Axiom 7 (First Distribution). $c(\vec{u}+\vec{v})=c\vec{u}+c\vec{v}$.

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Second Distribution”,“label”:“second-distribution”,“_index”:7}] Axiom 8 (Second Distribution). $(c+d)\vec{u}=c\vec{u}+d\vec{u}$.

Examples §

• vector spaces over $\mathbb{R}$
  • $V={\mathbb{R}}^2$
  • $V=\{f:\mathbb{R}\to \mathbb{R}\}$
  • $V=\{\text{continuous }f:\mathbb{R}\to \mathbb{R}\}$
  • $V=\{p(x)=a_0+a_{1}x+\cdots +a_n{x}^n\}$ where $a_0,\dots a_n\in \mathbb{R}$
  • For field $K$, $K^n=\{(a_1,a_2,\dots ,a_n)\}|a_i\in K$
    • $K-$vector space

Subspace §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Vector Subspace”,“label”:“vector-subspace”,“_index”:8}] Definition 9 (Vector Subspace). If $V$ is a $K-$vector subspace then a subspace of $V$ is a subset $W\subset V$ s.t. $W$ is also a $K-$vector space using the same operations.

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Closure under Operation”,“label”:“closure-under-operation”,“_index”:9}] Definition 10 (Closure under Operation). A subset $S\subset V$ is closed under $\star$ if given ${\vec{s}}_1,{\vec{s}}_2\in S$ then ${\vec{s}}_1\star \overrightarrow{s_2}\in S$.

• vector spaces are also closed under addition and scalar multiplication, but didn’t have to define since addition and multiplication are binary operations and closing by default
• notice that

[!math|{“type”:“proposition”,“number”:“auto”,“setAsNoteMathLink”:false,“_index”:10}] Proposition 11. Nonempty $W\subset V$ is subspace $⟺$ $W$ is closed under addition and scalar multiplication

Axioms §

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“_index”:11}] Axiom 12. $\vec{0}\in W$ always.

\begin{proof}For any $\vec{w}\in W$, $0\cdot \vec{w}=\vec{0}⟹\vec{w}\in W$.\end{proof}

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“_index”:12}] Axiom 13. Additive Inverse $\in W$ always

\begin{proof}For any $\vec{w}\in W$, $-1\cdot \vec{w}=-\vec{w}$. \end{proof}

Subspaces of ${\mathbb{R}}^2$ §

• if $W$ is subspace, then
• $\vec{w}\in W\ne \vec{0}⟹\lambda \vec{w}\in W$ for $\lambda \in \mathbb{R}$
• $L=\{\lambda \cdot \vec{w}\mid \lambda \in \mathbb{R}\}$ is subspace (check for closure)
  • $L$ is actually just a line through $\vec{0}$ if visualized on plane

[!math|{“type”:“claim”,“number”:“auto”,“setAsNoteMathLink”:false,“_index”:13}] Claim 14. If $W\subset V$ is a subspace, it’s either $\vec{0},{\mathbb{R}}^2$, or line through $\vec{0}$.

\begin{proof}Given $W$, assume it’s not $\{\vec{0}\}$ \end{proof}

[!math|{“type”:“claim”,“number”:“auto”,“setAsNoteMathLink”:false,“_index”:14}] Claim 15. Given any $\vec{u}\in {\mathbb{R}}^2$, can write $\vec{u}=\lambda \vec{w}+\mu \vec{v}$ for unique $\lambda ,\mu$