Difference between revisions of "12-240/Classnotes for Tuesday September 25"
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Today's class dealt with the properties of vector spaces. | 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 <math>\in\!\,</math> V} | ||
+ | |||
+ | FxV={(c,v): c <math>\in\!\,</math> F, v <math>\in\!\,</math> V} | ||
+ | |||
+ | Then, (+): VxV->V is (v,w)= v+w; (.): FxV->V is (c,v)=cv | ||
+ | |||
+ | Such that | ||
+ | |||
+ | VS1 <math>\forall\!\,</math> x, y <math>\in\!\,</math> V: x+y = y+x | ||
+ | |||
+ | VS2 <math>\forall\!\,</math> x, y, z <math>\in\!\,</math> V: x+(y+z) = (x+y)+z | ||
+ | |||
+ | VS3 <math>\forall\!\,</math> x <math>\in\!\,</math> V: 0 ( of V) +x = x | ||
+ | |||
+ | VS4 <math>\forall\!\,</math> x <math>\in\!\,</math> V, <math>\exists \!\,</math> V <math>\in\!\,</math> V: v + x= 0 ( of V) | ||
+ | |||
+ | VS5 <math>\forall\!\,</math> x <math>\in\!\,</math> V, 1 (of F) .x = x | ||
+ | |||
+ | VS6 <math>\forall\!\,</math> a, b <math>\in\!\,</math> F, <math>\forall\!\,</math> x <math>\in\!\,</math> V: (ab)x = a(bx) | ||
+ | |||
+ | VS7 <math>\forall\!\,</math> a <math>\in\!\,</math> F, <math>\forall\!\,</math> x, y <math>\in\!\,</math> V: a(x + y)= ax + ay | ||
+ | |||
+ | VS8 <math>\forall\!\,</math> a, b <math>\in\!\,</math> F, <math>\forall\!\,</math> x <math>\in\!\,</math> 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 [[User:Richardm|Richardm]]== | ||
+ | <gallery> | ||
+ | Image:12-240-0925-1vectorspaces.jpg | ||
+ | Image:12-240-0925-2vectorspaces.jpg | ||
+ | Image:12-240-0925-3vectorspaces.jpg | ||
+ | Image:12-240-0925-4vectorspaces.jpg | ||
+ | Image:12-240-0925-5vectorspaces.jpg | ||
+ | Image:12-240-0925-6vectorspaces.jpg | ||
+ | </gallery> |
Latest revision as of 05:41, 7 December 2012
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Today's class dealt with the properties of vector spaces.
Contents |
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