09-240/Classnotes for Thursday September 24

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Convention for today: $x,y,z\in \mathbf {V}$ $a,b,c\in F$

A V.S. over $F:V,0,+,\cdot$ s.t.

VS1 $\forall x,y\in \mathbf {V} ,x+y=y+x$
VS2 $\ldots (x+y)+z=x+(y+z)$
VS3 $\ldots x+0=x$
VS4 $\forall x,\exists y{\mbox{ s.t. }}x+y=0$
VS5 $1\cdot x=x$
VS6 $a\cdot (b\cdot x)=(a\cdot b)\cdot x$
VS7 $a\cdot (x+y)=ax+ay$
VS8 $(a+b)\cdot x=ax+bx$

Examples

1. $\left\{{\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}:a_{i}\in F\right\}=F^{n}$
2. $\left\{{\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &\ddots &\vdots \\a_{m1}&\cdots &a_{mn}\end{pmatrix}}\right\}=\mathrm {M} _{m\times n}(F)$
3. Let S be a set (F is some field)

   ${\mathcal {F}}(S,F)=\{f:S\rightarrow F\}$

   S = Primary colours = {red, green, blue}

   F = F₂ = {0, 1}

   ${\mathcal {F}}(S,F)=\left\{{\begin{matrix}f_{1}(red)=0&f_{1}(green)=1&f_{1}(blue)=0\\\cdots \\f_{2}{\begin{pmatrix}{\mbox{red}}\\{\mbox{green}}\\{\mbox{blue}}\end{pmatrix}}={\begin{pmatrix}0\\1\\0\end{pmatrix}}&\cdots \end{matrix}}\right\}$

---

$S=\mathbb {N} =\{1,2,3,4,\ldots \}F=\mathbb {R}$

${\mathcal {F}}(\mathbb {N} ,\mathbb {R} )=\left\{{\begin{matrix}1&2&3&4&\ldots \\6&6&6&6&\ldots \\\pi &2\pi &e&62&\ldots \\\end{matrix}}\right\}=\{{\mbox{sequences}}\}$

$S={\begin{pmatrix}\vdots \\\vdots \end{pmatrix}}n\Rightarrow {\mathcal {F}}(S,F)=F^{n}$

1. $O_{{\mathcal {F}}(S,F)}(\sigma )=0_{F}\forall \sigma \in S$
2. $f,g\in {\mathcal {F}}(S,F)$
   $\,\!(f+g)(\sigma )=f(\sigma )+g(\sigma )$
   $f\in {\mathcal {F}}(S,F)$
   $a\in F\Rightarrow \forall \sigma \in S,S(af)(\sigma )=a\cdot (f(\sigma ))$

Claim: + is associative. Given $f,g,h\in {\mathcal {F}}(S,F),(f+g)+h=f+(g+h)\forall \sigma$

$((f+g)+h)(\sigma )=(f+g)(\sigma )+h(\sigma )$

$=(f(\sigma )+g(\sigma ))+h(\sigma )$

$=f(\sigma )+g(\sigma )+h(\sigma ){\mbox{ (by F2)}}$

$(f+(g+h))(\sigma )=f(\sigma )+(g+h)(\sigma )$

$=f(\sigma )+(g(\sigma )+h(\sigma ))$

$=f(\sigma )+g(\sigma )+h(\sigma )$

4. $\mathbb {C} {\mbox{ is a V.S. over }}\mathbb {R}$
5. $\mathbb {R} {\mbox{ is a V.S. over }}\mathbb {Q}$
6. $\mathbb {R} {\mbox{ is a V.S. over }}\mathbb {R}$
7. $\,\!\{0\}{\mbox{ is a V.S. over }}F$

Dull theorem

1. Cancellation: $x+y=x+z\Rightarrow y=z$ (add w to both sides s.t. x + w = 0)
2. 0 is unique
3. Negatives are unique: $x+y=0=x+z\Rightarrow y=z$
4. $0x=0.a\cdot 0=0$
5. $(-a)\cdot x=a\cdot (-x)=-(ax)$