12-240/Classnotes for Thursday September 27: Difference between revisions
| (5 intermediate revisions by 4 users not shown) | |||
| Line 60: | Line 60: | ||
=== |
=== Hints for proofs === |
||
1.a Same as for fields |
1.a Same as for fields |
||
| Line 72: | Line 72: | ||
3. Same as F |
3. Same as F |
||
4. 0x + 0x = (0+0)x |
4. 0_F + 0_F = 0_F => by [VS8]: 0x + 0x = (0+0)x = 0x = 0x + 0 [VS3] = 0 + 0x [VS1] |
||
| ⚫ | |||
| ⚫ | |||
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 |
||
| Line 83: | 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 == |
||
| Line 90: | 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 ≠ |
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 |
||
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> |
|||
Latest revision as of 14:57, 30 September 2012
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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
(Quick recap)
VS1. x + y = y + x
VS2. (x + y) + z = x + (y + z)
VS3. 0 vector
VS4. + inverse → -
VS5. 1x = x
VS6. a(bx) = (ab)x
VS7. a(x + y) = ax + ay
VS8. (a+b)x = ax + bx
Theorems
1.a x + z = y + z ⇒ x = y
1.b ax = ay, a ≠ 0, ⇒ x = y
1.c ax = bx, x ≠ 0, ⇒ a = b
2. 0 is unique.
3. Additive inverse is unique.
4. 0_F ∙ x = 0_V
5. a ∙ 0_V = 0_V
6. (-a) x = -(ax) = a(-x)
7. cx = 0 ⇔ c = 0 or x = 0_V
Hints for proofs
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.c Discussed after proof of 7, harder than you think at first glance.
2. Same as F.
3. Same as F
4. 0_F + 0_F = 0_F => by [VS8]: 0x + 0x = (0+0)x = 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 Add (-bx) to each side, use VS8 then VS6 -> (a-b)x =0, use property 7.
Subspaces
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.
1. ∀ x, y ∈ W, x + y ∈ W
2. ∀ c ∈ F, ∀ x ∈ W, cx ∈ W
Scanned notes upload by Starash
Page 1
Page 2
