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
|
![09-240-ClassPhoto.jpg](/images/thumb/6/6f/09-240-ClassPhoto.jpg/180px-09-240-ClassPhoto.jpg) Add your name / see who's in!
|
|
WARNING: The notes below, written for students and by students, are provided "as is", with absolutely no warranty. They can not be assumed to be complete, correct, reliable or relevant. If you don't like them, don't read them. It is a bad idea to stop taking your own notes thinking that these notes can be a total replacement - there's nothing like one's own handwriting!
Visit this pages' history tab to see who added what and when.
The real numbers A set
with two binary operators and two special elements
s.t.
![{\displaystyle F1.\quad \forall a,b\in \mathbb {R} ,a+b=b+a{\mbox{ and }}a\cdot b=b\cdot a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89e2084a598d3aee71bf052e635228ff28680dde)
![{\displaystyle F2.\quad \forall a,b,c,(a+b)+c=a+(b+c){\mbox{ and }}(a\cdot b)\cdot c=a\cdot (b\cdot c)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7a5a7306b8734d61df7726f4e4b61779dc48022)
![{\displaystyle {\mbox{(So for any real numbers }}a_{1},a_{2},...,a_{n},{\mbox{ one can sum them in any order and achieve the same result.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7834e3c93dc288e9cbddb9a82ee0359267eb3b8e)
![{\displaystyle F3.\quad \forall a,a+0=a{\mbox{ and }}a\cdot 1=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a082b5651b55efd71b8c5dad41c1ca9a89e52a55)
![{\displaystyle F4.\quad \forall a,\exists b,a+b=0{\mbox{ and }}\forall a\neq 0,\exists b,a\cdot b=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c28670c5b19bcc977e870f9b1eb5dadf80219396)
![{\displaystyle {\mbox{So }}a+(-a)=0{\mbox{ and }}a\cdot a^{-1}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/001dc0d80a90e25b173645ead18e028aa45db4dd)
![{\displaystyle {\mbox{(So }}(a+b)\cdot (a-b)=a^{2}-b^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4dbaa0a51225955499a55d55a1a3b46579353a)
![{\displaystyle \forall a,\exists x,x\cdot x=a{\mbox{ or }}a+x\cdot x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d71a04dc0d2158e1688436802e87c744d5a4a1c)
- Note: or means inclusive or in math.
![{\displaystyle F5.\quad (a+b)\cdot c=a\cdot c+b\cdot c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21604a4fa96fe8e1edba621cc64a185f452fedea)
Definition: A field is a set F with two binary operators
: F×F → F,
: F×F → F and two elements
s.t.
![{\displaystyle F1\quad {\mbox{Commutativity }}a+b=b+a{\mbox{ and }}a\cdot b=b\cdot a\forall a,b\in F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93874137d44d34260a9f9fcbe7fe7d523dda1a86)
![{\displaystyle F2\quad {\mbox{Associativity }}(a+b)+c=a+(b+c){\mbox{ and }}(a\cdot b)\cdot c=a\cdot (b\cdot c)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2fe1a974b35f76aec8419cbe96228452778168e)
![{\displaystyle F3\quad a+0=a,a\cdot 1=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/038718cdb35f50ca32b3f80148bdb18d771fece8)
![{\displaystyle F4\quad \forall a,\exists b,a+b=0{\mbox{ and }}\forall a\neq 0,\exists b,a\cdot b=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b3fec383fede7d5334150f44d93f4a502330a3c)
![{\displaystyle F5\quad {\mbox{Distributivity }}(a+b)\cdot c=a\cdot c+b\cdot c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0aa3a67ba8c2a33052b2319934b954ab4c640f1)
Examples
![{\displaystyle F=\mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/153faf37401acde5841386b70649e8616aabdea0)
![{\displaystyle F=\mathbb {Q} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e178128ef93304af241f3706aacc15908eb9bfb3)
![{\displaystyle i={\sqrt {-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/370c8cebe9634fbfc84c29ea61680b0ad4a1ae0d)
![{\displaystyle \,\!(a+bi)+(c+di)=(a+c)+(b+d)i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b426dc2b841f290fb62b08b29a7f78c4a0a0daa7)
![{\displaystyle \,\!0=0+0i,1=1+0i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/293b5598eb3af046e8bed474b5c5fb8155fd8d05)
![{\displaystyle \,\!F_{2}=\{0,1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d33ac4cd92e1a50e706b4713830b0e0af70b5258)
![{\displaystyle \,\!F_{7}=\{0,1,2,3,4,5,6\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca480aae37fc5ace8a4d80c553b95e7e904a1aa)
is not a field because not every element has a multiplicative inverse.
- Let
![{\displaystyle a=2.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/393128c6a62f3c6c70a643fd904ff6851ed32a34)
- Then
![{\displaystyle a\cdot 0=0,a\cdot 1=2,a\cdot 3=0,a\cdot 4=2,a\cdot 5=4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d0bc7f3f2fcd00c32dab006cbc4c0e9213fda2a)
- Therefore F4 fails; there is no number b in F6 s.t. a · b = 1
|
|
Ex. 5
+ |
0 |
1 |
2 |
3 |
4 |
5 |
6
|
0
|
0 |
1 |
2 |
3 |
4 |
5 |
6
|
1
|
1 |
2 |
3 |
4 |
5 |
6 |
0
|
2
|
2 |
3 |
4 |
5 |
6 |
0 |
1
|
3
|
3 |
4 |
5 |
6 |
0 |
1 |
2
|
4
|
4 |
5 |
6 |
0 |
1 |
2 |
3
|
5
|
5 |
6 |
0 |
1 |
2 |
3 |
4
|
6
|
6 |
0 |
1 |
2 |
3 |
4 |
5
|
|
Ex. 5
× |
0 |
1 |
2 |
3 |
4 |
5 |
6
|
0
|
0 |
0 |
0 |
0 |
0 |
0 |
0
|
1
|
0 |
1 |
2 |
3 |
4 |
5 |
6
|
2
|
0 |
2 |
4 |
6 |
1 |
3 |
5
|
3
|
0 |
3 |
6 |
2 |
5 |
1 |
4
|
4
|
0 |
4 |
1 |
5 |
2 |
6 |
3
|
5
|
0 |
5 |
3 |
1 |
6 |
4 |
2
|
6
|
0 |
6 |
5 |
4 |
3 |
2 |
1
|
|
Theorem: F2 is a field.
In order to prove that the associative property holds, make a table (similar to a truth table) for a, b and c.
a |
b |
c |
|
0 |
0 |
0 |
|
0 |
0 |
1 |
|
0 |
1 |
0 |
|
0 |
1 |
1 |
(0 + 1) + 1 =? 0 + (1 + 1) 1 + 1 =? 0 + 0 0 = 0
|
1 |
0 |
0 |
|
1 |
0 |
1 |
|
1 |
1 |
0 |
|
1 |
1 |
1 |
|
Theorem:
for
is a field iff (if and only if)
is a prime number
Proof:
Given a finite set with
elements in
, an element
will have a multiplicative inverse iff
This can be shown using Bézout's identity:
![{\displaystyle \exists x,y{\mbox{ s.t. }}ax+my=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc1e29a78c024de5bcd9981411d521cb2538b2e3)
![{\displaystyle \left(ax+my\right){\pmod {m}}=1{\pmod {m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5564b40ce4e6104f2a83f64fcd9b0322f7fb209)
![{\displaystyle ax=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3570e5a911900f1a9cc990737bafd321f3fd48)
![{\displaystyle x=a^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22c1cc14849e3718adce2c58916c83b294abe502)
We have shown that
has a multiplicative inverse if
and
are relatively prime. It is therefore a natural conclusion that if
is prime all elements in the set will satisfy
Multiplication is repeated addition.
One may interpret this as counting the units in a 23×27 rectangle; one may choose to count along either 23 rows or 27 columns, but both ways lead to the same answer.
You may also think of it as 27-n=23 23*23 + 23*n = 27*23.
Exponentiation is repeated multiplication, but it does not have the same properties as multiplication; 23 = 8, but 32 = 9.
Tedious Theorem
"cancellation property"
- Proof:
- By F4,
![{\displaystyle \exists d{\mbox{ s.t. }}b+d=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23c78a46113e4287d59923b9ee90331d67b2dd81)
![{\displaystyle \,\!(a+b)+d=(c+b)+d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7af7e347780e5723f2833b619c3060024d5eb2)
by F2
by choice of d
by F3
![{\displaystyle a\cdot b=c\cdot b,(b\neq 0)\Rightarrow a=c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ced49d865d0432d4f4d17f3e4da7be8d9ad620)
- Proof:
![{\displaystyle \,\!a+O'=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53eb7d4c3e3cac0c990094f4747d7e5ff5b43aa3)
by F3
by adding the additive inverse of a to both sides
![{\displaystyle a\cdot l'=a,a\neq 0\Rightarrow l'=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8583051031c201818cef594c353bc7ba112f27cb)
![{\displaystyle a+b=0=a+b'\Rightarrow b=b'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a58da83bf3225ca9dcb62b8096044292d68fc9ce)
![{\displaystyle \,\!{\mbox{Aside: }}a-b=a+(-b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/947118df0c750d982c13a051edd614576d2d07b8)
![{\displaystyle {\frac {a}{b}}=a\cdot b^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cdbee908aed375b5c831185e8683fb0d65d5f2c)
![{\displaystyle \,\!-(-a)=a,(a^{-1})^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1442200fb2fa12a2804046007cc9bed861baeabe)
- Proof:
by F3
by F5
![{\displaystyle =0=a\cdot 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9343fc806ed69a139ba635a6e4e5ecdc104dd792)
- So there is no 0−1
![{\displaystyle (-a)\cdot b=a\cdot (-b)=-(a\cdot b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d16ba907d21442887ae310ae701e8c09885466)
![{\displaystyle (-a)\cdot (-b)=a\cdot b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/181dae08317a84c955fc60ee081b2e7fe8b63f9c)
- (Bonus)
![{\displaystyle \,\!(a+b)(a-b)=a^{2}-b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9fa725ed61f67fa1ab6f9606809c4c3d5eb9fb0)
Quotation of the Day
......