09-240/Classnotes for Tuesday September 22

09-240/Navigation Panel   Additions to the MAT 240 web site no longer count towards good deed points # Week of... Notes and Links 1 Sep 7 Tue, About, Thu 2 Sep 14 Tue, HW1, HW1 Solution, Thu 3 Sep 21 Tue, HW2, HW2 Solution, Thu, Photo 4 Sep 28 Tue, HW3, HW3 Solution, Thu 5 Oct 5 Tue, HW4, HW4 Solution, Thu, 6 Oct 12 Tue, Thu 7 Oct 19 Tue, HW5, HW5 Solution, Term Test on Thu 8 Oct 26 Tue, Why LinAlg?, HW6, HW6 Solution, Thu 9 Nov 2 Tue, MIT LinAlg, Thu 10 Nov 9 Tue, HW7, HW7 Solution Thu 11 Nov 16 Tue, HW8, HW8 Solution, Thu 12 Nov 23 Tue, HW9, HW9 Solution, Thu 13 Nov 30 Tue, On the final, Thu S Dec 7 Office Hours F Dec 14 Final on Dec 16 To Do List The Algebra Song! Register of Good Deeds Misplaced Material Add your name / see who's in!

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

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Vectors:

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

Let F be a field. A vector space V over the field F is a set V (of vectors) with a special element 0_V, a binary operation + : V × V → V, a binary operation • : F × V → V.

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

   ...

3. [math]\displaystyle{ \mathcal F(S, F) }[/math]
4. Polynomials
5. [math]\displaystyle{ ... }[/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, 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.