12-240/Classnotes for Tuesday September 25 - Revision history

Peterlue: /* Theorem */

2012-12-07T09:41:28Z

Theorem

2012-12-07T09:40:39Z

Theorem

2012-12-07T09:39:31Z

Polynomials

← Older revision		Revision as of 05:39, 7 December 2012
Line 37:		Line 37:

	==Polynomials==		==Polynomials==
	'''Definition''' : Pn(F) = {all polynomials of degree less than or equal n with coefficients in F}		'''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}		= {anx^n + an-1x^n-1 + ... + a1x^1 + a1x^1}

2012-12-07T09:38:50Z

Definition

← Older revision		Revision as of 05:38, 7 December 2012
Line 5:		Line 5:

	== Definition ==		== Definition ==
	Let F is a field, a vector space V over F is a set V of vectors with special element O ( of V) and ~~tow~~ operations: (+): VxV->V, (.): FxV->V		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}		VxV={(v,w): v,w <math>\in\!\,</math> V}
Line 30:		Line 30:

	VS8 <math>\forall\!\,</math> a, b <math>\in\!\,</math> F, <math>\forall\!\,</math> x <math>\in\!\,</math> V: (a + b)x = ax + bx		VS8 <math>\forall\!\,</math> a, b <math>\in\!\,</math> F, <math>\forall\!\,</math> x <math>\in\!\,</math> V: (a + b)x = ax + bx


	== Examples ==		== Examples ==

2012-11-02T18:49:36Z

← Older revision		Revision as of 14:49, 2 November 2012
Line 37:		Line 37:
	==Properties ==		==Properties ==

			==Polynomials==
			'''Definition''' : Pn(F) = {all polynomials of degree less than or equal 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]]==		==Scanned Notes by [[User:Richardm\|Richardm]]==

2012-09-25T23:44:16Z

← Older revision		Revision as of 19:44, 25 September 2012
Line 30:		Line 30:

	VS8 <math>\forall\!\,</math> a, b <math>\in\!\,</math> F, <math>\forall\!\,</math> x <math>\in\!\,</math> V: (a + b)x = ax + bx		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 ==


	==Scanned Notes by [[User:Richardm\|Richardm]]==		==Scanned Notes by [[User:Richardm\|Richardm]]==

2012-09-25T23:43:11Z

Definition

← Older revision		Revision as of 19:43, 25 September 2012
Line 27:		Line 27:
	VS6 <math>\forall\!\,</math> a, b <math>\in\!\,</math> F, <math>\forall\!\,</math> x <math>\in\!\,</math> V: (ab)x = a(bx)		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		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

	==Scanned Notes by [[User:Richardm\|Richardm]]==		==Scanned Notes by [[User:Richardm\|Richardm]]==

2012-09-25T23:41:48Z

Definition

← Older revision		Revision as of 19:41, 25 September 2012
Line 23:		Line 23:
	VS4 <math>\forall\!\,</math> x <math>\in\!\,</math> V, <math>\exists \!\,</math> V <math>\in\!\,</math> V: v + x= 0 ( of V)		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
	VS5

	VS6
			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

	==Scanned Notes by [[User:Richardm\|Richardm]]==		==Scanned Notes by [[User:Richardm\|Richardm]]==

2012-09-25T23:38:52Z

Definition

← Older revision		Revision as of 19:38, 25 September 2012
Line 21:		Line 21:
	VS3 <math>\forall\!\,</math> x <math>\in\!\,</math> V: 0 ( of V) +x = x		VS3 <math>\forall\!\,</math> x <math>\in\!\,</math> V: 0 ( of V) +x = x

	VS4 <math>\forall\!\,</math> x <math>\in\!\,</math> V, <math>\~~exist~~\!\</math> V <math>\in\!\,</math> V: v + x= 0 ( of V)		VS4 <math>\forall\!\,</math> x <math>\in\!\,</math> V, <math>\exists \!\,</math> V <math>\in\!\,</math> V: v + x= 0 ( of V)

	VS5		VS5

2012-09-25T23:37:55Z

Definition

← Older revision		Revision as of 19:37, 25 September 2012
Line 21:		Line 21:
	VS3 <math>\forall\!\,</math> x <math>\in\!\,</math> V: 0 ( of V) +x = x		VS3 <math>\forall\!\,</math> x <math>\in\!\,</math> V: 0 ( of V) +x = x

			VS4 <math>\forall\!\,</math> x <math>\in\!\,</math> V, <math>\exist\!\</math> V <math>\in\!\,</math> V: v + x= 0 ( of V)
	VS4

	VS5		VS5
	VS6		VS6