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. $\mathrm M_{m \times n}(F)$

   ...

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