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 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 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 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 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} (of vectors) with a special element 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 0_V} , a binary operation 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 \times \mathbf V \rightarrow \mathbf V} , a binary operation 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 \cdot : \mathcal F \times \mathbf V \rightarrow \mathbf V} .

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

VS1 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 \forall x, y \in \mathbf V, x + y = y + x}
VS2 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 \cdots (x + y) + z = x + (y + z)}
VS3 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 \cdots x + 0 = x}
VS4 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 \forall x, \exists y \mbox{ s.t. } x + y = 0}
VS5 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 1 \cdot x = x}
VS6 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 \cdot (b \cdot x) = (a \cdot b) \cdot x}
VS7 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 \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

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)}

   ...

3. 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(S, F)}
4. Polynomials
5. 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 ...}

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.