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Week of...
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Notes and Links
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1
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Sep 10
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About This Class, Tuesday, Thursday
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2
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Sep 17
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HW1, Tuesday, Thursday, HW1 Solutions
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3
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Sep 24
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HW2, Tuesday, Class Photo, Thursday
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4
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Oct 1
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HW3, Tuesday, Thursday
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5
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Oct 8
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HW4, Tuesday, Thursday
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6
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Oct 15
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Tuesday, Thursday
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7
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Oct 22
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HW5, Tuesday, Term Test was on Thursday. HW5 Solutions
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8
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Oct 29
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Why LinAlg?, HW6, Tuesday, Thursday, Nov 4 is the last day to drop this class
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9
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Nov 5
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Tuesday, Thursday
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10
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Nov 12
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Monday-Tuesday is UofT November break, HW7, Thursday
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11
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Nov 19
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HW8, Tuesday,Thursday
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12
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Nov 26
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HW9, Tuesday , Thursday
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13
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Dec 3
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Tuesday UofT Fall Semester ends Wednesday
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F
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Dec 10
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The Final Exam (time, place, style, office hours times)
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Register of Good Deeds
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Add your name / see who's in!
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In the second day of the class, the professor continues on the definition of a field.
Definition of a field
Combined with a part from the first class, we have a complete definition as follow:
A field is a set "F' with two binary operations +,x defind on it, and two special elements 0 ≠ 1 such that
F1: commutative law
a, b F: a+b=b+a and a.b=b.a
F2: associative law
a, b, c F: (a+b)+c=a+(b+c) and (a.b).c= a.(b.c)
F3: the existence of identity elements
a , a+0=a and a.1=a
F4: existence of inverses
a F \0, c, d F such that a+c=o and a.d=1
F5: contributive law
a, b, c F, a.(b+c)=a.b + a.c
Theorems
Theorem 1: Cancellation laws
a, b, c F
if a+c=b+c, then a=b
if a.c=b.c and c0, then a=b
Theorem 2: Identity uniqueness
Identity elements 0 and 1 mentioned in F3 are unique
a, b, b' F
if a+b=a and a+b'=a, then b=b'=0
if a.b=a and a.b'=a and a0, then b=b'=1
Theorem 3: Inverse uniqueness
Elements c and d mentioned in F4 are unique
a, b, b' F
if a+b=0 and a+b'=0, then b=b'
if a.b=1 and a.b'=1, then b=b'
Theorem 4
a F
-( -a) = a
Theorem 4
a F
0.a= 0
Significance
inverse uniqueness
It makes sense to define an operation
-: F -> F called "negation"
For a F define -a to be equal that b F for which a+b=0, i.e, a+(-a)=0
Ex: F(5)={0,1,2,3,4}, define +,x
Question 1: What is(-3)?
Answer: -3=2 and -3 is unique
Similarly, the inverse uniqueness also makes sense a^(-1)
identity uniqueness
Lecture Notes, upload by Starash