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1617-257/Classnotes for Friday October 14

2016-10-20T02:50:42Z

Hameeral: Write up the proof of symmetry of second partial derivatives.

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== Symmetry of Second Partial Derivatives ==

'''(Note: This is based off of the proof in the textbook, and may be slightly different from how it was presented in lecture.)'''

=== Theorem ===

Let <math>A</math> be an open subset of <math>\mathbb{R}^2</math>, and let <math>f: A \rightarrow \mathbb{R}</math> be of class <math>C^2</math>. Then, for all <math>c \in A</math>, we have that <math>D_1 D_2 f(c) = D_2 D_1 f(c)</math>.

=== Proof ===

==== Step 1 ====

We begin by defining a function that will help us in our proof. Let <math>(a, b) \in A \subseteq \mathbb{R}^2</math> be an arbitrary point. We then define the function <math>\lambda</math> as follows:

<math>\lambda(h, k) = f(a + h, b + k) - f(a + h, b) - f(a, b + k) + f(a, b)</math>.

Why do we define such a function? In fact, we can define both of the partial derivatives in question in terms of <math>\lambda</math>:

<math>
\begin{align}
D_1 D_2 f(a, b) & = \frac{\partial}{\partial x} \frac{\partial f}{\partial y} (a, b) \\
& = \lim_{h \to 0} (\frac{1}{h}) [\frac{\partial f}{\partial y} (a + h, b)] - [\frac{\partial f}{\partial y} (a, b)] \\
& = \lim_{h \to 0} (\frac{1}{h}) [\lim_{k \to 0} (\frac{1}{k}) f(a + h, b + k) - f(a + h, b)] - [\lim_{k \to 0} (\frac{1}{k}) f(a, b + k) - f(a, b)] \\
& = \lim_{h \to 0} (\frac{1}{h}) [\lim_{k \to 0} (\frac{1}{k}) f(a + h, b + k) - f(a + h, b) - f(a, b + k) + f(a, b)] \\
& = \lim_{h \to 0} \lim_{k \to 0} (\frac{1}{hk}) [f(a + h, b + k) - f(a + h, b) - f(a, b + k) + f(a, b)] \\
& = \lim_{h \to 0} \lim_{k \to 0} (\frac{1}{hk}) \lambda(h, k)
\end{align}
</math>

And similarly for <math>D_2 D_1 f(a, b)</math>, but with the two limits taken in the opposite order.

==== Step 2 ====

Now, let's make use of this function. Let us consider the square <math>Q = [a, a + h] \times [b, b + k]</math>, where <math>h</math> and <math>k</math> are so small that <math>Q \subseteq A</math>. We will show that there exist points <math>p, q \in Q</math> such that:

<math>
\begin{align}
& \lambda(h, k) = D_2 D_1 f(p) \cdot hk \\
& \lambda(h, k) = D_1 D_2 f(q) \cdot hk
\end{align}
</math>

The proofs of the two equalities are symmetric, and thus we only explicitly prove the first one. We will prove this through a double application of the mean value theorem.

Consider the function <math>\phi</math>, defined such that <math>\phi(s) = f(s, b + k) - f(s, b)</math>. Then <math>\phi(a + h) - \phi(a) = \lambda(h, k)</math>. By hypothesis, <math>D_1 f</math> exists at all points of <math>A</math>, so we can differentiate <math>\phi</math> with respect to the first variable of <math>f</math> on the interval <math>[a, a + h]</math>. By the mean value theorem, this means that there exists a point <math>s_0 \in [a, a + h]</math> such that <math>\lambda(h, k) = \phi(a + h) - \phi(a) = \phi^\prime(s_0) \cdot h = [D_1 f(s_0, b + k) - D_1 f(s_0, b)] \cdot h</math>.

Now we want to apply the mean value theorem one more time. Consider the function <math>\gamma</math>, defined such that <math>\gamma(t) = D_1 f(s_0, t)</math>. By hypothesis, <math>D_2 D_1 f</math> exists at all points of <math>A</math>, so we can differentiate <math>\gamma</math> with respect to the second variable of <math>f</math> on the interval <math>[b, b + k]</math>. So, by the mean value theorem, there exists a point <math>t_0 \in [b, b + k]</math> such that <math>\gamma(b + k) - \gamma(b) = \gamma^\prime(t_0) \cdot k</math>. Thus:

<math>
\begin{align}
\lambda(h, k) & = [D_1 f(s_0, b + k) - D_1 f(s_0, b)] \cdot h \\
& = [\gamma(b + k) - \gamma(b)] \cdot h \\
& = \gamma^\prime(t_0) \cdot hk \\
& = D_2 D_1 f(s_0, t_0) \cdot hk
\end{align}
</math>

==== Step 3 ====

Now we can prove the theorem. Let <math>c = (a, b) \in A</math>, and let <math>t > 0</math> be so small that <math>Q_t = [a, a + t] \times [b, b + t] \subseteq A</math>. By what we have just shown, <math>\lambda(t, t) = D_2 D_1 f(p_t) \cdot t^2</math>, for some <math>p_t \in Q_t</math>.

<math>t</math> is the length of the sides of the rectangle <math>Q_t</math>, so as <math>t \rightarrow 0</math>, <math>p_t \rightarrow c</math>. By hypothesis, <math>D_2 D_1 f</math> is continuous, and so as <math>p_t \rightarrow c</math>, <math>D_2 D_1 f(p_t) \rightarrow D_2 D_1 f(c)</math>. Thus:

<math>\lim_{t \to 0} \frac{\lambda(t, t)}{t^2} = D_2 D_1 f(c)</math>.

We can use the same argument, and the second equality from step 2, to show that:

<math>\lim_{t \to 0} \frac{\lambda(t, t)}{t^2} = D_1 D_2 f(c)</math>.

Therefore, by the uniqueness of limits, the two quantities must be equal. <math>\square</math>

== Handwritten Lecture Notes in PDF ==
[[Media:1617-257(lecture14).PDF|MAT257 - Lecture14 (Oct 14)]]

1617-257/Navigation

2016-09-20T04:05:52Z

Hameeral: Move extra problems to their lecture pages

<noinclude>Back to [[1617-257]].<br/></noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center style="color: red;"
|colspan=3|'''Welcome to Math 257'''

<span style="font-size: 250%">Class Moved!</span>

As of now yet only until the end of this semester, we meet at [http://osm.utoronto.ca/class_spec/f?p=210:1:918950691886201::NO::: MC 254]!
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|Sep 12
|[[1617-257/About This Class|About This Class]]; Day 1 Handout ({{pensieve link|Classes/1617-257-AnalysisII/160912/Day1.pdf|pdf}}, {{pensieve link|Classes/1617-257-AnalysisII/160912/Day1.html|html}}); [[1617-257/Classnotes for Monday September 12|Monday]], [[1617-257/Classnotes for Wednesday September 14|Wednesday]], [[1617-257/Classnotes for Friday September 16|Friday]], Day 2 Handout ({{pensieve link|Classes/1617-257-AnalysisII/160914/Day2.pdf|pdf}}, {{pensieve link|Classes/1617-257-AnalysisII/160914/Day2.html|html}}); [[Media:1617-257_Notes_First week.pdf|First Week Notes]].
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|Sep 19
|[[1617-257/Classnotes for Monday September 19|Monday]], [[1617-257/Classnotes for Wednesday September 21|Wednesday]], [[1617-257/Classnotes for Friday September 23|Friday]], [[1617-257/Homework Assignment 1|HW1]].
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|Sep 26
|[[1617-257/Classnotes for Monday September 26|Monday]], [[1617-257/Classnotes for Wednesday September 28|Wednesday]], [[1617-257/Classnotes for Friday September 30|Friday]], [[1617-257/Homework Assignment 2|HW2]].
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|Oct 3
|[[1617-257/Classnotes for Monday October 3|Monday]], [[1617-257/Classnotes for Wednesday October 5|Wednesday]], [[1617-257/Classnotes for Friday October 7|Friday]], [[1617-257/Homework Assignment 3|HW3]].
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|Oct 10
|Monday is Thanksgiving, no class; [[1617-257/Homework Assignment 4|HW4]].
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|Oct 17
|[[1617-257/Homework Assignment 5|HW5]].
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|Oct 24
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|Oct 31
|Monday is last day to switch to MAT 235; Term test 1 on Tuesday at 5-7PM; [[1617-257/Homework Assignment 6|HW6]].
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|Nov 7
|Monday is last day to switch to MAT 237; Monday-Tuesday is UofT Fall Break; [[1617-257/Homework Assignment 7|HW7]].
|- align=left
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|Nov 14
|[[1617-257/Homework Assignment 8|HW8]].
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|Nov 21
|[[1617-257/Homework Assignment 9|HW9]].
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|Nov 28
|[[1617-257/Homework Assignment 10|HW10]].
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|Dec 5
|Semester ends on Wednesday - no class Friday.
|- align=left
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|Dec 12,19,26, Jan 2
|No classes: other classes' finals, winter break.
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|Jan 9
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|Jan 16
|Term test 2 on Tuesday at 5-7PM; [[1617-257/Homework Assignment 11|HW11]].
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|Jan 23
|[[1617-257/Homework Assignment 12|HW12]].
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|Jan 30
|[[1617-257/Homework Assignment 13|HW13]].
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|Feb 6
|[[1617-257/Homework Assignment 14|HW14]]; UofT examination table posted on Friday.
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|Feb 13
|[[1617-257/Homework Assignment 15|HW15]].
|- align=left
|align=center|R
|Feb 20
|Reading week - no classes.
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|Feb 27
|[[1617-257/Homework Assignment 16|HW16]].
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|Mar 6
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|Mar 13
|Term test 3 on Tuesday at 5-7PM; [[1617-257/Homework Assignment 17|HW17]].
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|Mar 20
|[[1617-257/Homework Assignment 18|HW18]].
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|Mar 27
|[[1617-257/Homework Assignment 19|HW19]].
|- align=left
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|Apr 3
|Semester ends on Wednesday - no class Friday.
|- align=left
|align=center|F
|Apr10-28
|[[1617-257/The Final Exam|The Final Exam]].
|- align=left
|colspan=3 align=center|[[1617-257/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:1617-257-ClassPhoto.jpg|310px|Class Photo]]<br/>[[1617-257/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|<math>\int_M d\omega=\int_{\partial M}\omega</math>
|}

1617-257/Classnotes for Wednesday September 14

2016-09-20T04:05:06Z

Hameeral:

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== Extra Problem ==

<gallery>
File:1617-256-Lecture_02_extra_problem.pdf|by [[User:Vlad|Vlad]]
</gallery>

1617-257/Classnotes for Friday September 16

2016-09-20T04:02:16Z

Hameeral:

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== Typeset Lecture Notes ==

<gallery>
File:Lecture_03.pdf|by [[User:Vlad|Vlad]]
</gallery>

== Handwritten Lecture Notes in PDF ==
[[Media:1617-257(lecture3).PDF|MAT257 - Lecture3 (Sep 16)]]

== Extra Problem ==

<gallery>
File:Mat257lec2answer.pdf|by [[User:Mohan Zhang|Mohan Zhang]]
</gallery>

1617-257/Navigation

2016-09-20T03:58:13Z

Hameeral: Move Friday lecture notes to their own page

<noinclude>Back to [[1617-257]].<br/></noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center style="color: red;"
|colspan=3|'''Welcome to Math 257'''

<span style="font-size: 250%">Class Moved!</span>

As of now yet only until the end of this semester, we meet at [http://osm.utoronto.ca/class_spec/f?p=210:1:918950691886201::NO::: MC 254]!
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
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|Sep 12
|[[1617-257/About This Class|About This Class]]; Day 1 Handout ({{pensieve link|Classes/1617-257-AnalysisII/160912/Day1.pdf|pdf}}, {{pensieve link|Classes/1617-257-AnalysisII/160912/Day1.html|html}}); [[1617-257/Classnotes for Monday September 12|Monday]], [[1617-257/Classnotes for Wednesday September 14|Wednesday]], [[1617-257/Classnotes for Friday September 16|Friday]], Day 2 Handout ({{pensieve link|Classes/1617-257-AnalysisII/160914/Day2.pdf|pdf}}, {{pensieve link|Classes/1617-257-AnalysisII/160914/Day2.html|html}}); [[Media:1617-257_Notes_First week.pdf|First Week Notes]]; [[Media:1617-256-Lecture_02_extra_problem.pdf|Lecture 2 Extra Problem]] ; [[Media:Mat257lec2answer.pdf|Lecture 3 Extra Problem]];.
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|Sep 19
|[[1617-257/Classnotes for Monday September 19|Monday]], [[1617-257/Classnotes for Wednesday September 21|Wednesday]], [[1617-257/Classnotes for Friday September 23|Friday]], [[1617-257/Homework Assignment 1|HW1]].
|- align=left
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|Sep 26
|[[1617-257/Classnotes for Monday September 26|Monday]], [[1617-257/Classnotes for Wednesday September 28|Wednesday]], [[1617-257/Classnotes for Friday September 30|Friday]], [[1617-257/Homework Assignment 2|HW2]].
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|Oct 3
|[[1617-257/Classnotes for Monday October 3|Monday]], [[1617-257/Classnotes for Wednesday October 5|Wednesday]], [[1617-257/Classnotes for Friday October 7|Friday]], [[1617-257/Homework Assignment 3|HW3]].
|- align=left
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|Oct 10
|Monday is Thanksgiving, no class; [[1617-257/Homework Assignment 4|HW4]].
|- align=left
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|Oct 17
|[[1617-257/Homework Assignment 5|HW5]].
|- align=left
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|Oct 24
|
|- align=left
|align=center|8
|Oct 31
|Monday is last day to switch to MAT 235; Term test 1 on Tuesday at 5-7PM; [[1617-257/Homework Assignment 6|HW6]].
|- align=left
|align=center|9
|Nov 7
|Monday is last day to switch to MAT 237; Monday-Tuesday is UofT Fall Break; [[1617-257/Homework Assignment 7|HW7]].
|- align=left
|align=center|10
|Nov 14
|[[1617-257/Homework Assignment 8|HW8]].
|- align=left
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|Nov 21
|[[1617-257/Homework Assignment 9|HW9]].
|- align=left
|align=center|12
|Nov 28
|[[1617-257/Homework Assignment 10|HW10]].
|- align=left
|align=center|13
|Dec 5
|Semester ends on Wednesday - no class Friday.
|- align=left
|align=center|B
|Dec 12,19,26, Jan 2
|No classes: other classes' finals, winter break.
|- align=left
|align=center|14
|Jan 9
|
|- align=left
|align=center|15
|Jan 16
|Term test 2 on Tuesday at 5-7PM; [[1617-257/Homework Assignment 11|HW11]].
|- align=left
|align=center|16
|Jan 23
|[[1617-257/Homework Assignment 12|HW12]].
|- align=left
|align=center|17
|Jan 30
|[[1617-257/Homework Assignment 13|HW13]].
|- align=left
|align=center|18
|Feb 6
|[[1617-257/Homework Assignment 14|HW14]]; UofT examination table posted on Friday.
|- align=left
|align=center|19
|Feb 13
|[[1617-257/Homework Assignment 15|HW15]].
|- align=left
|align=center|R
|Feb 20
|Reading week - no classes.
|- align=left
|align=center|20
|Feb 27
|[[1617-257/Homework Assignment 16|HW16]].
|- align=left
|align=center|21
|Mar 6
|
|- align=left
|align=center|22
|Mar 13
|Term test 3 on Tuesday at 5-7PM; [[1617-257/Homework Assignment 17|HW17]].
|- align=left
|align=center|23
|Mar 20
|[[1617-257/Homework Assignment 18|HW18]].
|- align=left
|align=center|24
|Mar 27
|[[1617-257/Homework Assignment 19|HW19]].
|- align=left
|align=center|25
|Apr 3
|Semester ends on Wednesday - no class Friday.
|- align=left
|align=center|F
|Apr10-28
|[[1617-257/The Final Exam|The Final Exam]].
|- align=left
|colspan=3 align=center|[[1617-257/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:1617-257-ClassPhoto.jpg|310px|Class Photo]]<br/>[[1617-257/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|<math>\int_M d\omega=\int_{\partial M}\omega</math>
|}

1617-257/Classnotes for Friday September 16

2016-09-20T03:57:27Z

Hameeral: Move lecture notes here from navigation bar

14-240/Classnotes for Wednesday October 8

2014-10-11T20:42:25Z

Hameeral:

{{14-240/Navigation}}

==Scanned Lecture Notes by [[User Yue.Jiang|Yue Jiang]]==

<gallery>
File:October 8 note 1.jpeg|page 1
File:October 8 note 2.jpeg|page 2
File:October 8 note 3.jpeg|page 3
File:4.jpg|page 4
File:5.jpg|page 5
</gallery>

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

<gallery>
File:MAT240 (Oct 8, 2014) 1 of 3.pdf|page 1
File:MAT240 (Oct 8, 2014) 2 of 3.pdf|page 2
File:MAT240 (Oct 8, 2014) 3 of 3.pdf|page 3
</gallery>

14-240/Classnotes for Wednesday October 1

2014-10-11T20:40:53Z

Hameeral:

{{14-240/Navigation}}
The "vitamins" slide we viewed today is {{Pensieve link|Classes/14-240/Vitamins.pdf|Vitamins.pdf}}.

{{14-240:Dror/Students Divider}}

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

<gallery>
File:MAT240 Oct 1 (1 of 4).pdf|page 1
File:MAT240 Oct 1 (2 of 4).pdf|page 2
File:MAT240 Oct 1 (3 of 4).pdf|page 3
File:MAT240 Oct 1 (4 of 4).pdf|page 4
</gallery>

14-240/Classnotes for Monday October 6

2014-10-11T20:38:57Z

Hameeral:

{{14-240/Navigation}}

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

<gallery>
File:MAT240 (Oct 6, 2014) 1 of 2.pdf|page 1
File:MAT240 (Oct 6, 2014) 2 of 2.pdf|page 2
</gallery>

14-240/Classnotes for Wednesday September 24

2014-09-29T01:03:28Z

Hameeral: Format scanned lecture notes into a gallery.

{{14-240/Navigation}}

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

<gallery>
File:MAT240 Sept 22 (1 of 4).pdf|page 1
File:MAT240 Sept 24 (2 of 4).pdf|page 2
File:MAT240 Sept 24 (3 of4).pdf|page 3
File:MAT240 Sept 24 (4 of 4).pdf|page 4
</gallery>

14-240/Classnotes for Monday September 22

2014-09-29T01:02:08Z

Hameeral: Format scanned lecture notes into a gallery.

{{14-240/Navigation}}
Polar coordinates:
* <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>

The Fundamantal Theorem of Algebra:
<math>a_n \times z^{n} + a_n-1 \times z^{n-1} + \dots + a_0</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.

* 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 <math>O_v \in V</math> and two binary operations:
* <math>+ : V \times V -> V</math>
* <math>\times : V \times V -> V</math>

s.t.
* <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_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_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_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>.

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

<gallery>
File:MAT240 Sept 22,14 (1 of 2).pdf|page 1
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</gallery>

14-240/Classnotes for Wednesday September 17

2014-09-29T01:00:26Z

Hameeral: Format scanned lecture notes into a gallery.

{{14-240/Navigation}}
Today's handout was {{Pensieve link|Classes/14-240/nb/TheComplexField.pdf|TheComplexField.pdf}}.

{{14-240:Dror/Students Divider}}

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

<gallery>
File:MAT240 Sept 17 (1 of 2).pdf|page 1
File:MAT240 Sept 17 (2 of 2).pdf|page 2
</gallery>

14-240/Classnotes for Monday September 15

2014-09-29T00:58:45Z

Hameeral: Format scanned lecture notes into a gallery.

{{14-240/Navigation}}
==Definition of Subtraction and Division==
* Subtraction: if <math>a, b \in F, a - b = a + (-b)</math>.
* Division: if <math>a, b \in F, a / b = a \times b^{-1}</math>.

==Basic Properties of a Field (cont'd)==

'''8.''' <math>\forall a \in F</math>, <math>a \times 0 = 0</math>.
;Proof of 8
:By F3 , <math>a \times 0 = a \times (0 + 0)</math>
:By F5 , <math>a \times (0 + 0) = a \times 0 + a \times 0</math>;
:By F3 , <math>a \times 0 = 0 + a \times 0</math>;
:By Thm P1, <math>0 = a \times 0</math>.

'''9.''' <math>\nexists b \in F</math> s.t. <math>0 \times b = 1</math>;
:<math>\forall b \in F</math> s.t. <math>0 \times b \neq 1</math>.
;Proof of 9
:By F3 , <math>\times b = 0 \neq 1</math>.

'''10.''' <math>(-a) \times b = a \times (-b) = -(a \times b)</math>.

'''11.''' <math>(-a) \times (-b) = a \times b</math>.

'''12.''' <math>a \times b = 0 \iff a = 0</math> or <math>b = 0</math>.
;Proof of 12
:'''<= :'''
:By P8 , if <math>a = 0</math> , then <math>a \times b = 0 \times b = 0</math>;
:By P8 , if <math>b = 0</math> , then <math>a \times b = a \times 0 = 0</math>.
:'''=> :''' Assume <math>a \times b = 0 </math> , if a = 0 we are done;
:Otherwise , by P8 , <math>a \neq 0 </math> and we have <math>a \times b = 0 = a \times 0</math>;
:by cancellation (P2) , <math>b = 0</math>.

<math>(a + b) \times (a - b) = a^2 - b^2</math>.
;Proof
:By F5 , <math>(a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b))</math>
:<math>= a^2 - b^2</math>

==Theorem==
<math>\exists! \iota : \Z \rightarrow F</math> s.t.
:1. <math>\iota(0) = 0 , \iota(1) = 1</math>;
:2. <math>\forall m ,n \in \Z, \iota(m+n) = \iota(m) + \iota(n)</math>;
:3. <math>\forall m ,n \in \Z, \iota(m\times n) = \iota(m) \times \iota(n)</math>.

;Examples
<math>\iota(2) = \iota(1+1) = \iota(1) + \iota(1) = 1 + 1;</math>
<math>\iota(3) = \iota(2+1) = \iota(2) + \iota(1) = \iota(2) + 1;</math>

......

In F2:
<math>
\begin{align}
27 ----> \iota(27) &= \iota(26 + 1)\\
&= \iota(26) + \iota(1)\\
&= \iota(26) + 1\\
&= \iota(13 \times 2) + 1\\
&= \iota(2) \times \iota(13) + 1\\
&= (1 + 1) \times \iota(13) + 1\\
&= 0 \times \iota(13) + 1\\
&= 1
\end{align}
</math>

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

<gallery>
File:MAT 240 lecture 3 (1 of 2).pdf|page 1
File:MAT240 lectuire 3 (2 of 2).pdf|page 2
</gallery>

14-240/Classnotes for Monday September 15

2014-09-16T13:42:46Z

Hameeral: Fix some typesetting.

Definition:
Subtraction: if <math>a, b \in F, a - b = a + (-b)</math>.
Division: if <math>a, b \in F, a / b = a * b^{-1}</math>.

Theorem:

8. For every <math>a</math> belongs to F , <math>a * 0 = 0</math>.
proof of 8: By F3 , <math>a * 0 = a * (0 + 0)</math>;
By F5 , <math>a * (0 + 0) = a * 0 + a * 0</math>;
By F3 , <math>a * 0 = 0 + a * 0</math>;
By Thm P1 ,<math>0 = a * 0</math>.

9. There not exists <math>b</math> belongs to F s.t. <math>0 * b = 1</math>;
For every <math>b</math> belongs to F s.t. <math>0 * b </math>is not equal to <math>1</math>.
proof of 9: By F3 , <math>0 * b = 0 </math>is not equal to <math>1</math>.

10. <math>(-a) * b = a * (-b) = -(a * b)</math>.

11. <math>(-a) * (-b) = a * b</math>.

12. <math>a * b = 0 iff a = 0 or b = 0</math>.
proof of 12: <= : By P8 , if <math>a = 0</math> , then <math>a * b = 0 * b = 0</math>;
By P8 , if <math>b = 0</math> , then <math>a * b = a * 0 = 0</math>.
=> : Assume <math>a * b = 0 </math> , if a = 0 we have done;
Otherwise , by P8 , <math>a </math> is not equal to <math>0 </math>and we have <math>a * b = 0 = a * 0</math>;
by cancellation (P2) , <math>b = 0</math>.

<math>(a + b) * (a - b) = a^2 - b^2</math>.
proof: By F5 , <math>(a + b) * (a - b) = a * (a + (-b)) + b * (a + (-b))
= a * a + a * (-b) + b * a + (-b) * b
= a^2 - b^2</math>
Theorem :
There exists !(unique) iota <math>\iota : \Z \rightarrow F</math> s.t.
1. <math>\iota(0) = 0 , \iota(1) = 1</math>;
2. For every <math>m ,n</math> belong to <math>Z</math> , <math>\iota(m+n) = \iota(m) + \iota(n)</math>;
3. For every <math>m ,n</math> belong to <math>Z</math> , <math>\iota(m*n) = \iota(m) * \iota(n)</math>.

iota(2) = iota(1+1) = iota(1) + iota(1) = 1 + 1;
iota(3) = iota(2+1) = iota(2) + iota(1) = iota(2) + 1;
......

In F2 , <math>27 ----> iota(27) = iota(26 + 1)
= iota(26) + iota(1)
= iota(26) + 1
= iota(13 * 2) + 1
= iota(2) * iota(13) + 1
= (1 + 1) * iota(13) + 1
= 0 * iota(13) + 1
= 1</math>

14-240/Classnotes for Wednesday September 10

2014-09-14T16:11:11Z

Hameeral: Formatted the lecture note links into a gallery.

Knowledge about Fields:

During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.

===The Real Numbers===

====Properties of Real Numbers====

The real numbers are a set <math>\R</math> with two binary operations:

<math>+ : \R \times \R \rightarrow \R</math>

<math>* : \R \times \R \rightarrow \R</math>

such that the following properties hold.

R1 : <math>\forall a, b \in \R, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

R2 : <math>\forall a, b, c \in \R, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

R3 : <math>\forall a \in \R, a + 0 = a ~\&~ a * 1 = a</math> (existence of units: 0 is known as the "additive unit" and 1 as the "multiplicative unit")

R4 : <math>\forall a \in \R, \exists b \in \R, a + b = 0</math>;
<math>\forall a \in \R, a \ne 0 \Rightarrow \exists b \in \R, a * b = 1</math> (existence of inverses)

R5 : <math>\forall a, b, c \in \R, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)

====Properties That Do Not Follow from R1-R5====

There are properties which are true for <math>\R</math>, but do not follow from R1 to R5. For example ('''note''' that OR in mathematics denotes an "inclusive or"):
<math>\forall a \in \R, \exists x \in \R, a = x^2</math> OR <math>-a = x^2</math> (the existence of square roots)

Consider another set that satisfies all the properties R1 to R5. In <math>\Q</math> (the rational numbers), let us take </math>a = 2</math>. There is no <math>x \in \Q</math> such that <math>x^2 = a = 2</math>, so the statement above is not true for the rational numbers!

-------------------------------------------------------------------------------------------------------------------------------------------------------

===Fields===

====Definition====

A "field" is a set <math>\mathbb{F}</math> along with a pair of binary operations:
<math>+ : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

<math>* : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

and along with a pair <math>(0, 1) \in \mathbb{F}, 0 \ne 1</math>, such that the following properties hold.

F1 : <math>\forall a, b \in \mathbb{F}, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

F2 : <math>\forall a, b, c \in \mathbb{F}, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

F3 : <math>\forall a \in \mathbb{F}, a + 0 = a ~\&~ a * 1 = a</math> (existence of units)

F4 : <math>\forall a \in \mathbb{F}, \exists b \in \mathbb{F}, a + b = 0</math>;
<math>\forall a \in \mathbb{F}, a \ne 0 \Rightarrow \exists b \in \mathbb{F}, a * b = 1</math> (existence of inverses)

F5 : <math>\forall a, b, c \in \mathbb{F}, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)
====Examples====

# <math>\R</math> is a field.
# <math>\Q</math> (the rational numbers) is a field.
# <math>\C</math> (the complex numbers) is a field.
# <math>\mathbb{F} = \{0, 1\}</math> with operations defined as follows (known as <math>\mathbb{F}_2</math> or <math>\Z/2</math>) is a field:

{| class="wikitable"
|-
! +
! 0
! 1
|-
! 0
| 0
| 1
|-
! 1
| 1
| 0
|}

{| class="wikitable"
|-
! *
! 0
! 1
|-
! 0
| 0
| 0
|-
! 1
| 0
| 1
|}

<br style="clear:both;">

More generally, for every prime number <math>P</math>, <math>\mathbb{F}_p = \{0, 1, 2, 3, \cdots, p - 1\}</math> is a field, with operations defined by
<math>(a, b) \rightarrow a + b \mod P</math>.

An example: <math>\mathbb{F}_7 = \{0, 1, 2, 3, 4, 5, 6\}</math>, the operations are like remainders when divided by 7, or "like remainders mod 7". For example, <math>4 + 6 = 4 + 6 \mod 7</math> and <math>3 * 5 = 3 * 5 \mod 7</math>.

====Basic Properties of Fields====

'''Theorem''':
Let <math>\mathbb{F}</math> be a field, and let <math>a, b, c</math> denote elements of <math>\mathbb{F}</math>. Then:
# <math>a + b = c + b \Rightarrow a = c</math> (cancellation law)
# <math>b \ne 0 ~\&~ a * b = c * b \Rightarrow a = c</math>
Proof of 1:
1. By F4, <math>\exists b' \in \mathbb{F}, b + b' = 0</math>.
We know that <math>a + b = c + b</math>;
Therefore <math>(a + b) + b' = (c + b) + b'</math>.
2. By F2, <math>a + (b + b') = c + (b + b')</math>,
so by the choice of <math>b'</math>, we know that <math>a + 0 = c + 0</math>.
3. Therefore, by F3, <math>a = c</math>.
＾_＾

Proof of 2: more or less the same.

3. If <math>0' \in \mathbb{F}</math> is "like 0", then it is 0:
If <math>0' \in \mathbb{F}</math> satisfies <math>\forall a \in \mathbb{F}, a + 0' = a</math>, then 0' = 0.

4. If <math>1' \in \mathbb{F}</math> is "like 1", then it is 1:
If <math>1' \in \mathbb{F}</math> satisfies that <math>\forall a \in \mathbb{F}, a * 1' = a</math>, then 1' = 1.

Proof of 3 :
1. By F3 , 0' = 0' + 0.
2. By F1 , 0' + 0 = 0 + 0'.
3. By assumption on 0', 0' = 0 + 0' = 0.
＾_＾

5. <math>\forall a, b, b' \in \mathbb{F}, a + b = 0 ~\&~ a + b' = 0 \Rightarrow b = b'</math>:
In any field "<math>-a</math>" makes sense because it is unique -- it has an unambigous meaning.
<math>(-a):</math> the <math>b</math> such that <math>a + b = 0</math>.

6. <math>\forall a, b, b' \in \mathbb{F}, a \ne 0 ~\&~ a * b = 1 = a * b' \Rightarrow b = b'</math>:
In any field, if <math>a \ne 0</math>, "<math>a^{-1}</math>" makes sense.

Proof of 5 :
1. <math>a + b = 0 = a + b'</math>.
2. By F1, <math>b + a = b'+ a</math>.
3. By cancellation, <math>b = b'</math>.
＾_＾

7. <math>-(-a) = a</math> and when <math>a \ne 0</math>, <math>(a^{-1})^{-1} = a</math>.

Proof of 7 :
1. By definition, <math>a + (-a) = 0</math>. (*)
2. By definition, <math>(-a) + (-(-a) = 0</math>.
3. By (*) and F1, <math>(-a) + a = 0</math>.
4. By property 5, <math>-(-a) = a</math>.
＾_＾

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

<gallery>
File:MAT 240 (1 of 2) Sept 10, 2014.pdf‎|page 1
File:MAT 240 (2 of 2) Sept 10, 2014.pdf‎|page 2
</gallery>

14-240/Classnotes for Monday September 8

2014-09-14T16:02:49Z

Hameeral: Cleaned up and typeset Monday's notes.

{{14-240/Navigation}}
We went over "What is this class about?" ({{Pensieve link|Classes/14-240/one/What_is_This_Class_AboutQ.pdf|PDF}}, {{Pensieve link|Classes/14-240/What_is_This_Class_AboutQ.html|HTML}}), then over "[[14-240/About This Class|About This Class]]", and then over the first few properties of real numbers that we will care about.

{{14-240:Dror/Students Divider}}

===The Real Numbers===

The real numbers are a set <math>\R</math> with 2 binary operations + and *, defined as follows:

<math>+: \R \times \R \rightarrow \R</math>

<math>*: \R \times \R \rightarrow \R</math>

in addition to 2 special elements <math>0, 1 \in \R</math> such that <math>0 \ne 1</math>, with the following properties:

====The Commutative Law====

R1: For every <math>a, b \in \R</math>, we have:
<math>a + b = b + a</math> (commutative law for addition)
<math>ab = ba</math> (commutative law for multiplication)

====The Associative Law====

R2: For every <math>a, b, c \in \R</math>, we have:

<math>(a + b) + c = a + (b + c)</math>
<math>(ab)c = a(bc)</math>

This is not true for a number of other sets in our lives! For example, the associative law does not hold for the English language. Consider the phrase "pretty little girls": "(pretty little) girls" does not mean the same thing as "pretty (little girls)".
<math>(PL)G \ne P(LG)</math>

So the associative property does not hold for the English language.

====Existence of Units====

R3: For every <math>a \in \R</math>:

<math>a + 0 = a</math> (additive unit)
<math>a * 1 = a</math> (multiplicative unit)

== Wednesday September 10th 2014 - Fields ==

The real numbers: A set |R with +,x : |R x |R -> |R & <math>0=/=1</math> are elements of |R
such that

R1: For every <math>a, b</math> that are elements of |R , <math>a + b = b + a</math> and <math>ab = ba </math>

R2: For every <math>a, b, c</math> that are elements of |R, <math>( a + b ) + c = a + ( b + c )</math> and <math> (ab)c = a(bc)</math>

R3: For every <math>a</math> that is an element of |R, <math>a + 0 = a</math> and <math> a * 1 = a</math>

R4: For every a that is an element of |R there exists b that is an element of |R such that <math>a + b = 0</math>
& for every a that is an element of |R and <math>a =/= 0</math> there exists b that is an element |R such that <math>a * b = 1</math>

R5: For every <math>a, b, c</math> that are elements of |R, <math>( a + b ) c = ac + bc</math>

<math>( a + b ) * ( a - b ) = a^2 - b^2</math> follows from R1-R5

The following is true for the Real Numbers but does not follow from R1-R5
For every a that is an element of |R there exists an <math>x</math> that is an element of |R such that <math>a = x^2 or a + x^2 = 0</math>

However we can see that it does not follow from R1-R5 because we can find a field that obeys R1-R5 yet does not follow the above rule.
An example of this is the Rational Numbers |Q. In |Q take <math>a = 2</math> and there does not exist <math>x</math> such that <math>2 = x^2</math> or <math>2 + x^2 = 0</math>

The Definition Of A Field:
A "Field" is a set F along with a pair of binary operations +,x : FxF -> F and along with a pair <math>0, 1</math> that are elements of F such that <math>0 =/= 1</math> and such that R1-R5 hold.

R1: For every <math>a, b</math> that are elements of F , <math>a + b = b + a</math> and <math> ab = ba</math>

R2: For every <math>a, b, c</math> that are elements of F, <math>( a + b ) + c = a + ( b + c )</math> and <math>(ab)c = a(bc)</math>

R3: For every <math>a</math> that is an element of F, <math>a + 0 = a</math> and <math>a * 1 = a</math>

R4: For every <math>a</math> that is an element of F there exists <math>b</math> that is an element of F such that <math>a + b = 0</math>
& for every <math>a</math> that is an element of F and <math>a =/= 0</math> there exists <math>b</math> that is an element F such that <math>a * b = 1</math>

R5: For every <math>a, b, c</math> that are elements of F, <math>( a + b ) c = ac + bc</math>

Example

1. |R is a field (real numbers)
2. |Q is a field (rational numbers)
3. |C is a field (complex numbers)
4. F = {0, 1}

*insert table of addition and multiplication*

Proposition: F is a Field
checking F5

etc...

F = {0 , 1} = F2 = Z/2

Do the same for F7

*insert table of addition and multiplication*

"Like remainders when you divide by 7"
"like remainders mod 7'

Theorem (that shall remain unproved) :
For every prime number P, FP = {0 , 1 , 2 , ... , p-1 }
along with + & x defined as above
<math>( a , b ) -> a + b mod p</math>
is a field.

----

Theorem: (basic properties of Fields)

Let F be a Field, and let a , b , c denote elements of F
Then:
1. <math>a + b = c + b -> a = c </math>
"Cancellation" still holds
2. <math>b =/= 0 , ab = cb -> a = c</math>
3. If <math>0'</math> is an element of F and satisfies for every <math>a , a + 0' = a</math> , then <math>0' = 0 </math>
4. If <math>1'</math> is "like 1" then <math>1' = 1</math>

... to be continued...

14-240/Classnotes for Wednesday September 10

2014-09-13T21:47:39Z

Hameeral: Typeset the equations and added mathematical notation.

14-240/Navigation

2014-09-13T20:39:58Z

Hameeral:

<noinclude>Back to [[14-240]].<br/></noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center style="color: red;"
|colspan=3|'''Welcome to Math 240!'''
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 8
|[[14-240/About This Class|About This Class]], What is this class about? ({{Pensieve link|Classes/14-240/one/What_is_This_Class_AboutQ.pdf|PDF}}, {{Pensieve link|Classes/14-240/What_is_This_Class_AboutQ.html|HTML}}), [[14-240/Classnotes for Monday September 8|Monday]], [[14-240/Classnotes for Wednesday September 10|Wednesday]]
|- align=left
|align=center|2
|Sep 15
|[[14-240/Homework Assignment 1|HW1]]
|- align=left
|align=center|3
|Sep 22
|[[14-240/Homework Assignment 2|HW2]], [[14-240/Class Photo|Class Photo]]
|- align=left
|align=center|4
|Sep 29
|[[12-240/Homework Assignment 3|HW3]]
|- align=left
|align=center|5
|Oct 6
|[[14-240/Homework Assignment 4|HW4]]
|- align=left
|align=center|6
|Oct 13
|No Monday class (Thanksgiving)
|- align=left
|align=center|7
|Oct 20
|[[14-240/Homework Assignment 5|HW5]], [[14-240/Term Test|Term Test]] at tutorials on Tuesday
|- align=left
|align=center|8
|Oct 27
|[[14-240/Homework Assignment 6|HW6]]
|- align=left
|align=center|9
|Nov 3
|Monday is the last day to drop this class, [[14-240/Homework Assignment 7|HW7]]
|- align=left
|align=center|10
|Nov 10
|[[14-240/Homework Assignment 8|HW8]]
|- align=left
|align=center|11
|Nov 17
|Monday-Tuesday is UofT November break, [[14-240/Homework Assignment 9|HW9]]
|- align=left
|align=center|12
|Nov 24
|[[14-240/Homework Assignment 10|HW10]]
|- align=left
|align=center|13
|Dec 1
|Wednesday is a "makeup Monday"!
|- align=left
|align=center|F
|Dec 8
|[[14-240/The Final Exam|The Final Exam]]?
|- align=left
|align=center|F
|Dec 15
|[[14-240/The Final Exam|The Final Exam]]?
|- align=left
|colspan=3 align=center|[[14-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:14-240-ClassPhoto.jpg|310px|Class Photo]]<br/>[[14-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:14-240-Splash.png|310px]]
|}

14-240/About This Class

2014-09-13T20:37:20Z

Hameeral: /* The Term Test */

{{14-240/Navigation}}
===Crucial Information===
{{14-240/Crucial Information}}

'''URL:''' <span class="plainlinks">{{SERVER}}/drorbn/index.php?title=14-240</span>.

===Abstract===
Taken from the Faculty of Arts and Science [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Calendar]:
<blockquote>
A theoretical approach to: vector spaces over arbitrary fields including <math>{\mathbb C}</math>, <math>{\mathbb Z}_p</math>. Subspaces, bases and dimension. Linear transformations, matrices, change of basis, similarity, determinants. Polynomials over a field (including unique factorization, resultants). Eigenvalues, eigenvectors, characteristic polynomial, diagonalization. Minimal polynomial, Cayley-Hamilton theorem.
</blockquote>

*Prerequisite: high school level calculus.
*Co-requisite: MAT157Y1
*Distribution Requirement Status: This is a Science course
*Breadth Requirement: The Physical and Mathematical Universes (5)

[[Image:Friedberg_Insel_Spence_Cover.jpg|right|200px]]

===Text Book(s)===
Our main text book will be ''Linear Algebra'' (fourth edition) by Friedberg, Insel and Spence, ISBN 0-13-008451-4; it is a required reading.

===Wiki===
The class web site is a wiki, as in [http://www.wikipedia.org Wikipedia] - meaning that anyone can and is welcome to edit almost anything and in particular, students can post notes, comments, pictures, whatever. Some rules, though -
* This wiki is a part of my ([[User:Drorbn|Dror's]]) academic web page. All postings on it must be class-related (or related to one of the other projects I'm involved with).
* You must login to edit. To get an account, email me the class you are taking ([[14-240]]), your preferred login name, your real name and your email address if different from the address you are writing from.
* Criticism is fine, but no insults or foul language, please.
* I ([[User:Drorbn|Dror]]) will allow myself to exercise editorial control, when necessary.
* The titles of all pages related to this class should begin with "14-240/" or with "14-240-", just like the title of this page.
* For most 14-240 pages, it is a good idea to put a line containing only the string <tt><nowiki>{{14-240/Navigation}}</nowiki></tt> at the top of the page. This template inserts the class' "navigation panel" on the top right of the page.
* To edit the navigation panel itself, click on the word "Navigation" on the upper right of the panel. '''Use caution!''' Such edits affect many other pages! Note that due to page-caching, such edits take some time to propagate to the pages that include the navigation panel. To force immediate propagation to a given page, reload that page with the string "<tt>&action=purge</tt>" (meaning: "purge cached version") appended to the page's URL.
* Neatness matters! Material that is posted in an appealing manner will be read more, and thus will be more useful.
* Some further editing help is available at [[Help:Contents]].

===Marking Scheme===
There will be one term test (25% of the total grade) and a final exam (50%), as well as about 10 homework assignments (25%).

====The Term Test====
The term test will take place at tutorials on Tuesday October 21st, 1-3PM. A student who misses the term test without providing a valid reason (for example, a doctor’s note) within one week of the test will receive a mark of 0 on the term test. There will be no make-up term test. If a student misses the term test for a valid reason, the weight of the problem sets will increase to 35% and the weight of the final exam to 65%.

====Homework====
About 10 assignments will be posted on the course web page and distributed in class (usually on Mondays) approximately on the weeks shown in the class timeline. They will usually be due a week later at the tutorials (on Tuesdays) and they will be (at least partially) marked by the TAs. All students (including those who join the course late) will receive a mark of 0 on each assignment not handed in; though in computing the homework grade, your worst two assignments will not count. I encourage you to discuss the assignments with other students or browse the web, so long as you do at least some of the thinking on your own and you write up your own solutions. Remember that cheating is always possible and may increase your homework grade a bit. But it will hurt your appreciation of yourself, your knowledge and your exam grades a lot more.

===Good Deeds===
Students will be able to earn up to 25 "good deeds" points throughout the year for doing services to the class as a whole. There is no pre-set system for awarding these points, but the following will definitely count:
* Drawing a beautiful picture to illustrate a point discussed in class and posting it on this site.
* Taking class notes in nice handwriting, scanning them and posting them here.
* Typing up or formatting somebody else's class notes, correcting them or expanding them in any way.
* Writing an essay on expanding on anything mentioned in class and posting it here; correcting or expanding somebody else's article.
* Setup ''useful'' external resources: A web-based discussion forum? A Q/A site?
* Doing anything on our [[14-240/To do]] list.
* Any other service to the class as a whole.

Good deed points will count towards your final grade! If you got <math>n</math> of those, they are solidly yours and the above formula for the final grade will only be applied to the remaining <math>100-n</math> points. So if you got 25 good deed points (say) and your final grade is 80, I will report your grade as <math>25+80(100-25)/100=85</math>. Yet you can get an overall 100 even without doing a single good deed.

'''Important.''' For your good deeds to count, you '''must''' do them under your own name. So you must set up an account for yourself on this wiki and you must use it whenever you edit something. I will periodically check [[Special:Recentchanges|Recent changes]] to assign good deeds credits. Those credits will be made public (good deeds are public as a whole) towards the end of the course, at [[14-240/Register of Good Deeds]].

'''Important.''' The good deed points are an extra, a bonus, a treat. Very few will get many, and you should not count of them as a substitute for doing class work.

===Class Photo===
To help me learn your names, I will take a class photo on Wednesday of the third week of classes. I will post the picture on the class' web site and you will be ''required'' to send me an email and identify yourself in the picture or to identify yourself on the [[14-240/Class Photo|Class Photo]] page of this wiki.

===Accessibility Needs===
The University of Toronto is committed to accessibility. If you require accommodations for a disability, or have any accessibility concerns about the course, the classroom or course materials, please contact Accessibility Services as soon as possible: [mailto:disability.services@utoronto.ca disability.services@utoronto.ca] or http://studentlife.utoronto.ca/accessibility.

===Academic Integrity===
I have been asked to include with the course syllabus a link to the Office of Academic Integrity. Here it is: http://www.artsci.utoronto.ca/osai/students.

===How to Succeed in this Class===
* '''Keep up!''' Don't fall behind on reading, listening, and doing assignments! University goes at a different pace than high school. New material is covered once and just once. There will be no going over the same thing again and again - if you fall behind, you stay behind. Unless you are an Einstein, there is ''no way'' to do well in this class merely by attending lectures - you '''must''' think about the material more than 3 or 5 hours a week if you want it to sink in. And if you are planning on not attending lectures, well, think again. Most people find it very hard to pace their own studies without a human contact; if you'll try, you are likely to discover the hard way that you belong to the majority.
* If in high school you were the best in your class in math, now remember that everybody around you was the same. You may find that what was enough then simply doesn't cut it any more. Try to catch that early in the year!
* Math is about '''understanding''', not about memorizing. To understand is to internalize; it is to come to the point where whatever the professor does on the blackboard or whatever is printed in the books becomes '''yours'''; it is to come to the point where you appreciate why everything is done the way it is done, what does it mean, what are the reasons and motivations and what is it all good for. Don't settle for less!
* Keep asking yourself questions; many of them will be answered in class, but not all. Remember the old Chinese proverb:

<div style="text-align: center; font-size: 18px">'''"Teachers open the door, but you must enter by yourself"'''</div>
<div style="text-align: center; font-size: 18px">'''"师傅领进门,修行靠个人!"'''</div>