Additions to the MAT 240 web site no longer count towards good deed points

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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
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Class notes for today
Vectors:
 can be added
 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:
 $x,y,z\in \mathbf {V}$
 $a,b,c\in F$

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
 $F^{n}{\mbox{ for }}n\in \mathbb {N}$
 $F^{n}=\left\{{\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}:a_{i}\in F\right\}$
 ${\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}+{\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{pmatrix}}={\begin{pmatrix}a_{1}+b_{1}\\a_{2}+b_{2}\\\vdots \\a_{n}+b_{n}\end{pmatrix}}$
 $a{\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{pmatrix}}={\begin{pmatrix}ab_{1}\\ab_{2}\\\vdots \\ab_{n}\end{pmatrix}}$
 ...
 $\mathrm {M} _{m\times n}(F)$
 ...
 ${\mathcal {F}}(S,F)$
 Polynomials
 $...$
Food for thought
What is wrong with setting
${\begin{pmatrix}2&3\\4&5\\\end{pmatrix}}\cdot {\begin{pmatrix}6&7\\8&9\\\end{pmatrix}}={\begin{pmatrix}2\cdot 6&3\cdot 7\\4\cdot 8&5\cdot 9\\\end{pmatrix}}={\begin{pmatrix}12&21\\32&45\\\end{pmatrix}}?$
 Unnecessary for a V.S.
 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.