14-240/Classnotes for Monday September 22: Difference between revisions

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Polar coordinates:
Polar coordinates:
* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>
* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>
* <math>\r_1 \times e^{i\\theta_2} = r_1 \times (cos\theta + sin\theta</math>
* <math>r_1 \times e^{i\theta_2} = r_1 \times (cos\theta + sin\theta</math>


The Fundamantal Theorem of Algebra:
The Fundamantal Theorem of Algebra:
<math>\a_n \times z^{n} + \a_n-1 \times z^{n-1} + \dots + \a_0</math>
<math>a_n \times z^{n} + a_n-1 \times z^{n-1} + \dots + a_0</math>
where <math>\a_i \in C and \a_i != 0</math> has a soluion <math>z \in C</math>
where <math>a_i \in C </math>and<math> a_i != 0</math> has a soluion <math>z \in C</math>
In particular, <math>z^{2} - 1 = 0</math> has a solution.
In particular, <math>z^{2} - 1 = 0</math> has a solution.


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Definition of Vector Space:
Definition of Vector Space:
A "Vector Space" over a field F is a set V with a special element <math>\O_v \in V</math> and two binary operations:
A "Vector Space" over a field F is a set V with a special element <math>O_v \in V</math> and two binary operations:
* <math>+ : V \times V -> V</math>
* <math>+ : V \times V -> V</math>
* <math>\times : V \times V -> V</math>
* <math>\times : V \times V -> V</math>


s.t.
s.t.
* <math>\VS_1 : \forall x, y \in V, x + y = y + x</math>.
* <math>VS_1 : \forall x, y \in V, x + y = y + x</math>.
* <math>\VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z</math>.
* <math>VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z</math>.
* <math>\VS_3 : \forall x \in V, x + 0 = x</math>.
* <math>VS_3 : \forall x \in V, x + 0 = x</math>.
* <math>\VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.
* <math>VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.
* <math>\VS_5 : \forall x \in V, 1 \times x = x</math>.
* <math>VS_5 : \forall x \in V, 1 \times x = x</math>.
* <math>\VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.
* <math>VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.
* <math>\VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.
* <math>VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.
* <math>\VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.
* <math>VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.

==Scanned Lecture Notes by [[User:AM|AM]]==

<gallery>
File:MAT240 Sept 22,14 (1 of 2).pdf|page 1
File:MAT240 Sept 22,14 (2 of 2).pdf|page 2
</gallery>
==Scanned Lecture Notes by [[User Boyang.wu|Boyang.wu]]==

[[File:W31.pdf]]

Latest revision as of 00:57, 8 December 2014

DBN: Classes: 2014-15: 14-240 / Navigation
Welcome to Math 240!
(additions to this web site no longer count towards good deed points)
# Week of... Notes and Links
1 Sep 8 About This Class, What is this class about? (PDF, HTML), Monday, Wednesday
2 Sep 15 HW1, Monday, Wednesday, TheComplexField.pdf,HW1_solutions.pdf
3 Sep 22 HW2, Class Photo, Monday, Wednesday, HW2_solutions.pdf
4 Sep 29 HW3, Wednesday, Tutorial, HW3_solutions.pdf
5 Oct 6 HW4, Monday, Wednesday, Tutorial, HW4_solutions.pdf
6 Oct 13 No Monday class (Thanksgiving), Wednesday, Tutorial
7 Oct 20 HW5, Term Test at tutorials on Tuesday, Wednesday
8 Oct 27 HW6, Monday, Why LinAlg?, Wednesday, Tutorial
9 Nov 3 Monday is the last day to drop this class, HW7, Monday, Wednesday, Tutorial
10 Nov 10 HW8, Monday, Tutorial
11 Nov 17 Monday-Tuesday is UofT November break
12 Nov 24 HW9
13 Dec 1 Wednesday is a "makeup Monday"! End-of-Course Schedule, Tutorial
F Dec 8 The Final Exam
Register of Good Deeds
Class Photo
Add your name / see who's in!
14-240-Splash.png

Polar coordinates:

The Fundamantal Theorem of Algebra: where and has a soluion In particular, has a solution.


  • Forces can multiple by a "scalar"(number).

No "multiplication" of forces.


Definition of Vector Space: A "Vector Space" over a field F is a set V with a special element and two binary operations:

s.t.

  • .
  • .
  • .
  • .
  • .
  • .
  • .
  • .

Scanned Lecture Notes by AM

  • page 1

    page 1

  • page 2

    page 2

Scanned Lecture Notes by Boyang.wu

File:W31.pdf

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