12-240/Classnotes for Tuesday October 2 - Revision history

Peterlue: /* Subspace */

2012-12-07T09:43:04Z

Subspace

← Older revision		Revision as of 05:43, 7 December 2012
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	Proof:		Proof:

	First direction:		First direction "=>":

	if a non-empty subset W ⊂ V is a subspace , then W is a vector space over the operations of V .		if a non-empty subset W ⊂ V is a subspace , then W is a vector space over the operations of V .
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	Second direction		Second direction "<=":

	if a non-empty subset W ⊂ V is closed under the operations of V and contains 0 of V		if a non-empty subset W ⊂ V is closed under the operations of V and contains 0 of V

Dangkhoa.nguyen at 18:22, 4 October 2012

2012-10-04T18:22:44Z

← Older revision		Revision as of 14:22, 4 October 2012
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	VS3: As given, 0 of V <math>\in\!\,</math> W, pick any a in W ( possible since W is not empty)		VS3: As given, 0 of V <math>\in\!\,</math> W, pick any a in W ( possible since W is not empty)

	=> a <math>\in\!\,</math> V hence a + 0 = a		So, a <math>\in\!\,</math> V hence a + 0 = a

	Thus 0 is also additive identity element of W		Thus 0 is also additive identity element of W

Dangkhoa.nguyen at 18:20, 4 October 2012

2012-10-04T18:20:59Z

← Older revision		Revision as of 14:20, 4 October 2012
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	thus x + y = y + x ( and the sum <math>\in\!\,</math> W since W is closed under addition)		thus x + y = y + x ( and the sum <math>\in\!\,</math> W since W is closed under addition)

	VS2: (x + y) + z = x + (y + z) is proven similarly		VS2: (x + y) + z = x + (y + z) is proven similarly

	VS3: As given, 0 of V <math>\in\!\,</math> W, pick any a in W ( possible since W is not empty)		VS3: As given, 0 of V <math>\in\!\,</math> W, pick any a in W ( possible since W is not empty)

Dangkhoa.nguyen: /* Subspace */

2012-10-04T18:20:07Z

Subspace

← Older revision		Revision as of 14:20, 4 October 2012
Line 36:		Line 36:

	thus x + y = y + x ( and the sum <math>\in\!\,</math> W since W is closed under addition)		thus x + y = y + x ( and the sum <math>\in\!\,</math> W since W is closed under addition)
			VS2: (x + y) + z = x + (y + z) is proven similarly
			VS3: As given, 0 of V <math>\in\!\,</math> W, pick any a in W ( possible since W is not empty)

			=> a <math>\in\!\,</math> V hence a + 0 = a

			Thus 0 is also additive identity element of W

	== Class Notes ==		== Class Notes ==

Dangkhoa.nguyen: /* Subspace */

2012-10-04T17:04:33Z

Subspace

← Older revision		Revision as of 13:04, 4 October 2012
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	== Subspace ==		== Subspace ==

	Remind about the theorem of subspace: a non-empty subset W ⊂ V is a subspace iff is is closed under the operations of V and ~~contain~~ 0 of V		Remind about the theorem of subspace: a non-empty subset W ⊂ V is a subspace iff is is closed under the operations of V and contains 0 of V

	Proof:		Proof:
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	=> + W is closed under the operations of V.		=> + W is closed under the operations of V.

	+ W has a unique identity of addition: <math>\forall\!\,</math> a <math>\in\!\,</math> W: 0 + a = a		+ W has a unique identity of addition: <math>\forall\!\,</math> a <math>\in\!\,</math> W: 0 + a = a

	Moreover, a a <math>\in\!\,</math> V. Hence 0 is also identity of addtition of V		Moreover, a a <math>\in\!\,</math> V. Hence 0 is also identity of addtition of V
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	Second direction		Second direction

	if a non-empty subset W ⊂ V is closed under the operations of V		if a non-empty subset W ⊂ V is closed under the operations of V and contains 0 of V

	we need to prove that		we need to prove that W is a vector space over operations of V, hence, and subspace of V.

			Namely, we need to show that W satisfies all axioms of a vector space, but now we just consider some axioms and leave the rest to readers.

			VS1: Consider <math>\forall\!\,</math> x,y <math>\in\!\,</math> W => a,b <math>\in\!\,</math> V

			While V is a vector space

			thus x + y = y + x ( and the sum <math>\in\!\,</math> W since W is closed under addition)

	== Class Notes ==		== Class Notes ==

Dangkhoa.nguyen: /* Subspace */

2012-10-04T16:58:13Z

Subspace

← Older revision		Revision as of 12:58, 4 October 2012
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	First direction:		First direction:

	if a non-empty subset W ⊂ V is a subspace , then W is a vector space over the operations of V .		if a non-empty subset W ⊂ V is a subspace , then W is a vector space over the operations of V .

	=> + W is closed under the operations of V.		=> + W is closed under the operations of V.

	+ W has a unique identity of addition: <math>\forall\!\,</math> a <math>\in\!\,</math> W: 0 + a = a		+ W has a unique identity of addition: <math>\forall\!\,</math> a <math>\in\!\,</math> W: 0 + a = a

			Moreover, a a <math>\in\!\,</math> V. Hence 0 is also identity of addtition of V


	Second direction		Second direction

Dangkhoa.nguyen: /* Subspace */

2012-10-04T16:56:24Z

Subspace

← Older revision		Revision as of 12:56, 4 October 2012
Line 8:		Line 8:
	== Subspace ==		== Subspace ==

	Remind about the theorem of subspace: a non-empty subset W ⊂ V is a subspace iff is is closed under the operations of V		Remind about the theorem of subspace: a non-empty subset W ⊂ V is a subspace iff is is closed under the operations of V and contain 0 of V

	Proof: ~~First direction:~~		Proof:

			First direction:

			if a non-empty subset W ⊂ V is a subspace , then W is a vector space over the operations of V .

			=> + W is closed under the operations of V.

			+ W has a unique identity of addition: <math>\forall\!\,</math> a <math>\in\!\,</math> W: 0 + a = a

			Second direction

			if a non-empty subset W ⊂ V is closed under the operations of V

			we need to prove that

	== Class Notes ==		== Class Notes ==

Pratyush: /* Subspace */

2012-10-04T15:28:27Z

Subspace

← Older revision		Revision as of 11:28, 4 October 2012
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	== Subspace ==		== Subspace ==

	Remind about the theorem of subspace: a subset W ⊂ V is a subspace iff is is closed under the operations of V		Remind about the theorem of subspace: a non-empty subset W ⊂ V is a subspace iff is is closed under the operations of V

	Proof: First direction:		Proof: First direction:

Dangkhoa.nguyen: /* Class Notes */

2012-10-02T21:46:07Z

Class Notes

← Older revision		Revision as of 17:46, 2 October 2012
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	<gallery>		<gallery>
	~~Img~~:12-240-Oct-2-Page-1.jpg\|Page 1		Image:12-240-Oct-2-Page-1.jpg\|Page 1
	~~Img~~:12-240-Oct-2-Page-2.jpg\|Page 2		Image:12-240-Oct-2-Page-2.jpg\|Page 2
	~~Img~~:12-240-Oct-2-Page-3.jpg\|Page 3		Image:12-240-Oct-2-Page-3.jpg\|Page 3
	~~Img~~:12-240-Oct-2-Page-4.jpg\|Page 4		Image:12-240-Oct-2-Page-4.jpg\|Page 4
	~~Img~~:12-240-Oct-2-Page-5.jpg\|Page 5		Image:12-240-Oct-2-Page-5.jpg\|Page 5
	</gallery>		</gallery>

Dangkhoa.nguyen: /* Class Notes */

2012-10-02T21:45:35Z

Class Notes

← Older revision		Revision as of 17:45, 2 October 2012
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	<gallery>		<gallery>
	12-240-Oct-2-Page-1.jpg\|Page 1		Img:12-240-Oct-2-Page-1.jpg\|Page 1
	12-240-Oct-2-Page-2.jpg\|Page 2		Img:12-240-Oct-2-Page-2.jpg\|Page 2
	12-240-Oct-2-Page-3.jpg\|Page 3		Img:12-240-Oct-2-Page-3.jpg\|Page 3
	12-240-Oct-2-Page-4.jpg\|Page 4		Img:12-240-Oct-2-Page-4.jpg\|Page 4
	12-240-Oct-2-Page-5.jpg\|Page 5		Img:12-240-Oct-2-Page-5.jpg\|Page 5
	</gallery>		</gallery>

← Older revision		Revision as of 17:46, 2 October 2012
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	<gallery>		<gallery>
	~~Img~~:12-240-Oct-2-Page-1.jpg\|Page 1		Image:12-240-Oct-2-Page-1.jpg\|Page 1
	~~Img~~:12-240-Oct-2-Page-2.jpg\|Page 2		Image:12-240-Oct-2-Page-2.jpg\|Page 2
	~~Img~~:12-240-Oct-2-Page-3.jpg\|Page 3		Image:12-240-Oct-2-Page-3.jpg\|Page 3
	~~Img~~:12-240-Oct-2-Page-4.jpg\|Page 4		Image:12-240-Oct-2-Page-4.jpg\|Page 4
	~~Img~~:12-240-Oct-2-Page-5.jpg\|Page 5		Image:12-240-Oct-2-Page-5.jpg\|Page 5
	</gallery>		</gallery>

← Older revision		Revision as of 17:45, 2 October 2012
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	<gallery>		<gallery>
	12-240-Oct-2-Page-1.jpg\|Page 1		Img:12-240-Oct-2-Page-1.jpg\|Page 1
	12-240-Oct-2-Page-2.jpg\|Page 2		Img:12-240-Oct-2-Page-2.jpg\|Page 2
	12-240-Oct-2-Page-3.jpg\|Page 3		Img:12-240-Oct-2-Page-3.jpg\|Page 3
	12-240-Oct-2-Page-4.jpg\|Page 4		Img:12-240-Oct-2-Page-4.jpg\|Page 4
	12-240-Oct-2-Page-5.jpg\|Page 5		Img:12-240-Oct-2-Page-5.jpg\|Page 5
	</gallery>		</gallery>