09-240/Classnotes for Thursday September 24

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

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

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

Examples

1. ${\displaystyle \left\{{\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}:a_{i}\in F\right\}=F^{n}}$
2. ${\displaystyle \left\{{\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &\ddots &\vdots \\a_{m1}&\cdots &a_{mn}\end{pmatrix}}\right\}=\mathrm {M} _{m\times n}(F)}$
3. Let S be a set (F is some field)
${\displaystyle {\mathcal {F}}(S,F)=\{f:S\rightarrow F\}}$
S = Primary colours = {red, green, blue}
F = F2 = {0, 1}
${\displaystyle {\mathcal {F}}(S,F)=\left\{{\begin{matrix}f_{1}(red)=0&f_{1}(green)=1&f_{1}(blue)=0\\\cdots \\f_{2}{\begin{pmatrix}{\mbox{red}}\\{\mbox{green}}\\{\mbox{blue}}\end{pmatrix}}={\begin{pmatrix}0\\1\\0\end{pmatrix}}&\cdots \end{matrix}}\right\}}$

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

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

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

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

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

${\displaystyle ((f+g)+h)(\sigma )=(f+g)(\sigma )+h(\sigma )}$
${\displaystyle =(f(\sigma )+g(\sigma ))+h(\sigma )}$
${\displaystyle =f(\sigma )+g(\sigma )+h(\sigma ){\mbox{ (by F2)}}}$
${\displaystyle (f+(g+h))(\sigma )=f(\sigma )+(g+h)(\sigma )}$
${\displaystyle =f(\sigma )+(g(\sigma )+h(\sigma ))}$
${\displaystyle =f(\sigma )+g(\sigma )+h(\sigma )}$
1. ${\displaystyle \mathbb {C} {\mbox{ is a V.S. over }}\mathbb {R} }$
2. ${\displaystyle \mathbb {R} {\mbox{ is a V.S. over }}\mathbb {Q} }$
3. ${\displaystyle \mathbb {R} {\mbox{ is a V.S. over }}\mathbb {R} }$
4. ${\displaystyle \,\!\{0\}{\mbox{ is a V.S. over }}F}$

Dull theorem

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