09-240/Classnotes for Tuesday September 22

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Class notes for today

Vectors:

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 0V, a binary operation + : V × VV, a binary operation • : F × VV.

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

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. $\mathrm M_{m \times n}(F)$
...
3. $\mathcal F(S, F)$
4. Polynomials
5. $...$

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.