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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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Today's class dealt with the properties of vector spaces.
Definition
Let F be a field. A vector space V over F is a set V of vectors with special element O ( of V) and two operations: (+): VxV->V, (.): FxV->V
VxV={(v,w): v,w V}
FxV={(c,v): c F, v V}
Then, (+): VxV->V is (v,w)= v+w; (.): FxV->V is (c,v)=cv
Such that
VS1 x, y V: x+y = y+x
VS2 x, y, z V: x+(y+z) = (x+y)+z
VS3 x V: 0 ( of V) +x = x
VS4 x V, V V: v + x= 0 ( of V)
VS5 x V, 1 (of F) .x = x
VS6 a, b F, x V: (ab)x = a(bx)
VS7 a F, x, y V: a(x + y)= ax + ay
VS8 a, b F, x V: (a + b)x = ax + bx
Examples
Properties
Polynomials
Definition : Pn(F) = {all polynomials of degree less than or equal to n with coefficients in F}
= {anx^n + an-1x^n-1 + ... + a1x^1 + a1x^1}
0 = 0x^n + 0x^n-1 +...+ 0x^0
addition and multiplication: as you imagine
P(f) = {all polynomials with coefficients in F}
Take F= Z/2 |F| = 2
|P(F)| = infinite
in Pn(Z/2) x^3≠x^2
x^3 = 1*x^3+0x^2+0x+O = f
x^2 = 1*x^2+0x+0 = g
yet f(0)= g(0) and f(1)=g(1)
Theorem
1. Cancellation Laws
(a) x+z=y+z ==> x=y
(b) ax+ay,a≠0 ==> x=y
(c) x≠0 of V, ax=bx ==> a=b
2. 0 of V is unique
3. Negatives are unique (so subtraction makes sense
4.(0 of F)x = 0 of V
5. a*0=0
6. (-a)x= - (ax) = a(-x)
7. a*v=0 <==> a=0 or v=0
Proof
1. (a) x+z=y+z
Find a w s.t. z+w=0 (V.S. 4)
(x+z)+w = (y+z)+w
Use VS2
x+(z+w) = y +(z+w)
x + 0 = y + o
Use VS3 x=y
Scanned Notes by Richardm