12-240/Classnotes for Thursday September 27 - Revision history

Celtonmcgrath: /* Subspaces */

2012-09-30T19:57:28Z

Subspaces

← Older revision		Revision as of 15:57, 30 September 2012
Line 89:		Line 89:
	Definition: Let V be a vector space over a field F. A ''subspace'' W of V is a subset of V, has the operations inherited from V and 0_V of V, is itself a vector space.		Definition: Let V be a vector space over a field F. A ''subspace'' W of V is a subset of V, has the operations inherited from V and 0_V of V, is itself a vector space.

			=== Examples of subspaces ===
			Look at scanned notes for examples of subspaces!

	Theorem: A subset W ⊂ V, W ≠ ∅, is a subspace iff it is closed under the operations of V.		Theorem: A subset W ⊂ V, W ≠ ∅, is a subspace iff it is closed under the operations of V.

	1. ∀ x, y ∈ W, x + y ∈ W		1. ∀ x, y ∈ W, x + y ∈ W

2012-09-29T20:26:20Z

Subspaces

← Older revision		Revision as of 16:26, 29 September 2012
Line 90:		Line 90:


	Theorem: A subset W ⊂ V, W ≠ {∅}, is a subspace iff it is closed under the operations of V.		Theorem: A subset W ⊂ V, W ≠ ∅, is a subspace iff it is closed under the operations of V.

	1. ∀ x, y ∈ W, x + y ∈ W		1. ∀ x, y ∈ W, x + y ∈ W

2012-09-29T18:52:27Z

Theorems

2012-09-28T04:22:43Z

← Older revision		Revision as of 00:22, 28 September 2012
Line 95:		Line 95:

	2. ∀ c ∈ F, ∀ x ∈ W, cx ∈ W		2. ∀ c ∈ F, ∀ x ∈ W, cx ∈ W

			== Scanned notes upload by [[User:Starash\|Starash]] ==

			<gallery>
			Image:12-240-0927-1.jpg\|Page 1
			Image:12-240-0927-2.jpg\|Page 2
			</gallery>

2012-09-28T02:46:39Z

Rough sketches for proofs

← Older revision		Revision as of 22:46, 27 September 2012
Line 60:		Line 60:


	=== ~~Rough sketches~~ for proofs ===		=== Hints for proofs ===

	1.a Same as for fields		1.a Same as for fields
Line 82:		Line 82:
	To the right, Suppose c not= 0, then show x must equal 0.		To the right, Suppose c not= 0, then show x must equal 0.

			1.c Add (-bx) to each side, use VS8 then VS6 -> (a-b)x =0, use property 7.
	1.c

	== Subspaces ==		== Subspaces ==

2012-09-28T02:40:54Z

← Older revision		Revision as of 22:40, 27 September 2012
Line 73:		Line 73:

	4. 0x + 0x = (0+0)x [VS8] = 0x = 0x + 0 [VS3] = 0 + 0x [VS1]		4. 0x + 0x = (0+0)x [VS8] = 0x = 0x + 0 [VS3] = 0 + 0x [VS1]
		⚫	⇒ 0x + 0x = 0 + 0x ⇒ [Cancellation property] 0x = 0

⚫	=> 0x + 0x = 0 + 0x => [Cancellation property] 0x = 0

	5. Same as 4 except using 0_V + 0_V = 0_V and using VS7		5. Same as 4 except using 0_V + 0_V = 0_V and using VS7

2012-09-28T02:40:18Z

Rough sketches for proofs

2012-09-28T02:40:02Z

Rough sketches for proofs

← Older revision		Revision as of 22:40, 27 September 2012
Line 64:		Line 64:
	1.a Same as for fields		1.a Same as for fields

			1.b. Use similar proof as for fields, but use VS6 NOT F2b. F2b guarantees existence, but VS6 allows algebraic manipulation.
⚫	1.b

			1.c Discussed after proof of 7, harder than you think at first glance.

			2. Same as F.

			3. Same as F

			4. 0x + 0x = (0+0)x [VS8] = 0x = 0x + 0 [VS3] = 0 + 0x [VS1]
			=> 0x + 0x = 0 + 0x => [Cancellation property] 0x = 0

			5. Same as 4 except using 0_V + 0_V = 0_V and using VS7

			6. Skip

			7. Prove both ways: Easy way is to the left, show left is 0 if either on right is 0.
			To the right, Suppose c not= 0, then show x must equal 0.

		⚫	1.c

	== Subspaces ==		== Subspaces ==

2012-09-28T01:43:53Z

2012-09-28T01:42:23Z

← Older revision		Revision as of 21:42, 27 September 2012
Line 2:		Line 2:

	'''Vector Spaces'''		'''Vector Spaces'''

			== Reminders ==

			- Tag yourself in the photo!

			- Read along textbook 1.1 to 1.4

			- Riddle: Professor in ring with lion around the perimeter.
			Consider this: http://mathforum.org/library/drmath/view/63421.html

	== Vector space axioms ==		== Vector space axioms ==