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 [math]\displaystyle{ W=Z^2 }[/math] 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 [math]\displaystyle{ \mathcal F }[/math] be a field. A vector space [math]\displaystyle{ \mathbf V }[/math] over the field [math]\displaystyle{ \mathcal F }[/math] is a set [math]\displaystyle{ \mathbf V }[/math] (of vectors) with a special element [math]\displaystyle{ 0_V }[/math], a binary operation [math]\displaystyle{ + : \mathbf V \times \mathbf V \rightarrow \mathbf V }[/math], a binary operation [math]\displaystyle{ \cdot : \mathcal F \times \mathbf V \rightarrow \mathbf V }[/math].

Convention for today: [math]\displaystyle{ x, y, z \in \mathbf V }[/math] [math]\displaystyle{ a, b, c \in \mathcal F }[/math]

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

Proof of VS4

Take an arbitrary [math]\displaystyle{ x = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} \in F^n }[/math]

Set [math]\displaystyle{ y = \begin{pmatrix} -a_1 \\ -a_2 \\ \vdots \\ -a_n \end{pmatrix} }[/math] and note

[math]\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} }[/math]

Examples

1. [math]\displaystyle{ F^n \mbox{ for } n \in \mathbb N }[/math]

   [math]\displaystyle{ F^n = \left\{ \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} : a_i \in F \right\} }[/math]

   [math]\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} }[/math]

   [math]\displaystyle{ a \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix} = \begin{pmatrix} ab_1 \\ ab_2 \\ \vdots \\ ab_n \end{pmatrix} }[/math]

   ...

2. [math]\displaystyle{ \mathrm M_{m \times n}(F) }[/math]

   ...

Food for thought

What is wrong with setting

[math]\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} ? }[/math]

1. Unnecessary for a V.S.
2. This is useless