09-240/Classnotes for Tuesday September 22

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Some links

• The Complex Numbers by Computer.
• Dori Eldar's work on "mechanical computations": Machines as Calculating Devices and Computing the function $W=Z^{2}$ the hard way.
• The "Dimensions" video on "Nombres complexes", is at http://dimensions-math.org/Dim_reg_AM.htm (and then go to "Dimensions_5".

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Class notes for today

Vectors:

1. can be added
2. can be multiplied by a number (not another vector)

Let ${\mathcal {F}}$ be a field. A vector space $\mathbf {V}$ over the field ${\mathcal {F}}$ is a set $\mathbf {V}$ (of vectors) with a special element $0_{V}$, a binary operation $+:\mathbf {V} \times \mathbf {V} \rightarrow \mathbf {V}$, a binary operation $\cdot :{\mathcal {F}}\times \mathbf {V} \rightarrow \mathbf {V}$.

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

VS1 $\forall x,y\in \mathbf {V} ,x+y=y+x$
VS2 $\cdots (x+y)+z=x+(y+z)$
VS3 $\cdots x+0=x$
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 $(a+b)\cdot x=ax+bx$

Proof of VS4

Take an arbitrary $x={\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}\in F^{n}$

Set $y={\begin{pmatrix}-a_{1}\\-a_{2}\\\vdots \\-a_{n}\end{pmatrix}}$ and note

$x+y={\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}+{\begin{pmatrix}-a_{1}\\-a_{2}\\\vdots \\-a_{n}\end{pmatrix}}={\begin{pmatrix}a_{1}+(-a_{1})\\a_{2}+(-a_{2})\\\vdots \\a_{n}+(-a_{n})\end{pmatrix}}={\begin{pmatrix}0\\0\\\vdots \\0\end{pmatrix}}=0_{F^{n}}$

Examples

1. $F^{n}{\mbox{ for }}n\in \mathbb {N}$

   $F^{n}=\left\{{\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}:a_{i}\in F\right\}$

   ${\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}+{\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{pmatrix}}={\begin{pmatrix}a_{1}+b_{1}\\a_{2}+b_{2}\\\vdots \\a_{n}+b_{n}\end{pmatrix}}$

   $a{\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{pmatrix}}={\begin{pmatrix}ab_{1}\\ab_{2}\\\vdots \\ab_{n}\end{pmatrix}}$

   ...

2. $\mathrm {M} _{m\times n}(F)$

   ...

3. ${\mathcal {F}}(S,F)$
4. Polynomials
5. $...$

Food for thought

What is wrong with setting

${\begin{pmatrix}2&3\\4&5\\\end{pmatrix}}\cdot {\begin{pmatrix}6&7\\8&9\\\end{pmatrix}}={\begin{pmatrix}2\cdot 6&3\cdot 7\\4\cdot 8&5\cdot 9\\\end{pmatrix}}={\begin{pmatrix}12&21\\32&45\\\end{pmatrix}}?$

1. Unnecessary for a V.S.
2. This is useless, since it does not describe reality. For example, a mathematical theory with 46 dimensions can be perfect and mathematically elegant, but if the only solution to it is a universe in which life cannot form it is not reality, hence we have no use for it.