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

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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




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




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Theorem: A subset W C V, W != {emptyset}, 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. forall x, y elementof W, x + y elementof W
1. ∀ x, y ∈ W, x + y ∈ W


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

Revision as of 20:39, 27 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

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


Rough sketches for proofs

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.


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

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