12-240/Classnotes for Thursday September 27: Difference between revisions

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'''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 ==
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VS3. 0 vector
VS3. 0 vector


VS4. + inverse -> -
VS4. + inverse → -


VS5. 1x = x
VS5. 1x = x
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== Theorems ==
== Theorems ==


1.a x + z = y + z => x = y
1.a x + z = y + z ⇒ x = y


1.b ax = ay, a != 0, => x = y
1.b ax = ay, a ≠ 0, ⇒ x = y


1.c ax = bx, x != 0, => a = b
1.c ax = bx, x ≠ 0, ⇒ a = b




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4. 0_F * x = 0_V
4. 0_F ∙ x = 0_V




5. a * 0_V = 0_V
5. a ∙ 0_V = 0_V




Line 48: Line 57:




7. cx = 0 <=> c = 0 or x = 0_V
7. cx = 0 ⇔ c = 0 or x = 0_V




=== Rough sketches for proofs ===
=== 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 ==
== Subspaces ==
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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.

⚫
1. ∀ x, y ∈ W, x + y ∈ W


2. ∀ c ∈ F, ∀ x ∈ W, cx ∈ W
⚫
Theorem: A subset W C V, W != {emptyset}, is a subspace iff it is closed under the operations of V.


== Scanned notes upload by [[User:Starash|Starash]] ==
⚫
1. forall x, y elementof W, x + y elementof W


<gallery>
2. forall c elementof F, forall x elementof W, cx elementof W
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

DBN: Classes: 2012-13: 12-240 / Navigation
Additions to this web site no longer count towards good deed points.
# Week of... Notes and Links
1 Sep 10 About This Class, Tuesday, Thursday
2 Sep 17 HW1, Tuesday, Thursday, HW1 Solutions
3 Sep 24 HW2, Tuesday, Class Photo, Thursday
4 Oct 1 HW3, Tuesday, Thursday
5 Oct 8 HW4, Tuesday, Thursday
6 Oct 15 Tuesday, Thursday
7 Oct 22 HW5, Tuesday, Term Test was on Thursday. HW5 Solutions
8 Oct 29 Why LinAlg?, HW6, Tuesday, Thursday, Nov 4 is the last day to drop this class
9 Nov 5 Tuesday, Thursday
10 Nov 12 Monday-Tuesday is UofT November break, HW7, Thursday
11 Nov 19 HW8, Tuesday,Thursday
12 Nov 26 HW9, Tuesday , Thursday
13 Dec 3 Tuesday UofT Fall Semester ends Wednesday
F Dec 10 The Final Exam (time, place, style, office hours times)
Register of Good Deeds
12-240-ClassPhoto.jpg
Add your name / see who's in!
12-240-Splash.png

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

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    Page 1

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