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".

WARNING: The notes below, written for students and by students, are provided "as is", with absolutely no warranty. They can not be assumed to be complete, correct, reliable or relevant. If you don't like them, don't read them. It is a bad idea to stop taking your own notes thinking that these notes can be a total replacement - there's nothing like one's own handwriting! Visit this pages' history tab to see who added what and when.

Class notes for today

Vectors:

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

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} be a field. A vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} over the field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} \in F^n}

Set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix}} and note

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^n \mbox{ for } n \in \mathbb N}

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^n = \left\{ \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} : a_i \in F \right\}}

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix} = \begin{pmatrix} ab_1 \\ ab_2 \\ \vdots \\ ab_n \end{pmatrix}}

   ...

2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm M_{m \times n}(F)}

   ...

Food for thought

What is wrong with setting

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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