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06-240/Classnotes For Thursday November 23

2006-12-04T09:43:41Z

Gokmen:

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==Class Notes==

[[Media:MAT Lect019.pdf|Scan of Week 11 Lecture 2 notes]]

[[Media:06-240-lec noted nov23rd.pdf|Lecture_notes nov23rd]]

==Tutorial Notes==

[[Media:MAT Tut011.pdf|Scan of Week 11 Tutorial notes]]

File:06-240-lec noted nov23rd.pdf

2006-12-04T09:42:20Z

Gokmen:

06-240/Classnotes For Tuesday November 21

2006-12-04T09:30:54Z

Gokmen:

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==More about the [[User:Wongpak|Wongpak]] Matrices==

In [[Talk:06-240/Classnotes_For_Tuesday_November_14]], [[User:Wongpak]] asked something about row echelon form and reduced row echelon form, and gave the following matrices as specific examples:

{| align=center
|-
|<math>A_1=\begin{pmatrix}1&3&2&4&2\\0&1&2&3&4\\0&0&0&1&2\\0&0&0&0&0 \end{pmatrix}</math>
|width=10%|
|<math>A_2=\begin{pmatrix}1&0&-4&0&-6\\0&1&2&0&-2\\0&0&0&1&2\\0&0&0&0&0 \end{pmatrix}</math>
|}

So let us assume row reduction leads us to the systems <math>A_1x=b</math> or <math>A_2x=b</math>. What does it tell us about the solutions? Let us start from the second system:

{| align=center
|- align=center
|<math>\begin{pmatrix}1&0&-4&0&-6\\0&1&2&0&-2\\0&0&0&1&2\\0&0&0&0&0\end{pmatrix} \begin{pmatrix}x_1\\x_2\\x_3\\x_4\\x_5\end{pmatrix} = \begin{pmatrix}b_1\\b_2\\b_3\\b_4\end{pmatrix}</math>
|width=15%|or
|
{|
|-
|<math>x_1</math>
|
|<math>-4x_3</math>
|
|<math>-6x_5</math>
|<math>=</math>
|<math>b_1</math>
|-
|
|<math>x_2</math>
|<math>+2x_3</math>
|
|<math>-2x_5</math>
|<math>=</math>
|<math>b_2</math>
|-
|
|
|
|<math>x_4</math>
|<math>+2x_5</math>
|<math>=</math>
|<math>b_3</math>
|-
|
|
|
|
|align=center|<math>0</math>
|<math>=</math>
|<math>b_4</math>
|}
|}

Well, quite clearly if <math>b_4\neq 0</math> this system has no solutions, but if <math>b_4=0</math> it has solutions no matter what <math>b_1</math>, <math>b_2</math> and <math>b_3</math> are. Finally, for any given values of <math>b_1</math>, <math>b_2</math> and <math>b_3</math> we can choose the values of <math>x_3</math> and <math>x_5</math> (the variables corresponding the columns containing no pivots) as we please, and then get solutions by setting the "pivotal variables" in terms of the non-pivotal ones as follows: <math>x_1=b_1+4x_3+6x_5</math>, <math>x_2=b_2-2x_3+2x_5</math> and <math>x_4=b_3-2x_5</math>.

What about the system corresponding to <math>A_1</math>? It is
{| align=center
|- align=center
|<math>\begin{pmatrix}1&3&2&4&2\\0&1&2&3&4\\0&0&0&1&2\\0&0&0&0&0\end{pmatrix} \begin{pmatrix}x_1\\x_2\\x_3\\x_4\\x_5\end{pmatrix} = \begin{pmatrix}b_1\\b_2\\b_3\\b_4\end{pmatrix}</math>
|width=15%|or
|
{|
|- align=center
|<math>x_1</math>
|<math>+3x_2</math>
|<math>+2x_3</math>
|<math>+4x_4</math>
|<math>+2x_5</math>
|<math>=</math>
|<math>b_1</math>
|- align=center
|
|<math>x_2</math>
|<math>+2x_3</math>
|<math>+3x_4</math>
|<math>+4x_5</math>
|<math>=</math>
|<math>b_2</math>
|- align=center
|
|
|
|<math>x_4</math>
|<math>+2x_5</math>
|<math>=</math>
|<math>b_3</math>
|-
|
|
|
|
|align=center|<math>0</math>
|<math>=</math>
|<math>b_4</math>
|}
|}

Here too we have solutions iff <math>b_4=0</math>, and if <math>b_4=0</math>, we have the freedom to choose the non-pivotal variables <math>x_3</math> and <math>x_5</math> as we please. But now the formulas for fixing the pivotal variables <math>x_1</math>, <math>x_2</math> and <math>x_4</math> in terms of the non-pivotal ones are a bit harder.

==Class notes==

[[Media:MAT Lect018.pdf|Scan of Week 11 Lecture 1 notes]]

[[Media:06-240-lec notes nov21st.pdf|Lecture_notes november21st]]

06-240/Classnotes For Tuesday November 21

2006-12-04T09:30:17Z

Gokmen:

File:06-240-lec notes nov21st.pdf

2006-12-04T09:28:45Z

Gokmen:

06-240/Classnotes For Thursday November 16

2006-12-04T08:44:57Z

Gokmen:

{{06-240/Navigation}}

[[Media:06-240-nov16lec-1.jpeg|Nov16 Lecture notes: 1 of 2]]

[[Media:06-240-nov16lec-2.jpeg|Nov16 Lecture notes: 2 of 2]]

[[Media:06-240-nov16tut-1.jpeg|Nov16 Tutorial notes: 1 of 4]]

[[Media:06-240-nov16tut-2.jpeg|Nov16 Tutorial notes: 2 of 4]]

[[Media:06-240-nov16tut-3.jpeg|Nov16 Tutorial notes: 3 of 4]]

[[Media:06-240-nov16tut-4.jpeg|Nov16 Tutorial notes: 4 of 4]]

[[Media:MAT Lect017.pdf|Scan of Week 10 Lecture 2 notes]]

[[Media:MAT Tut010.pdf|Scan of Week 10 Tutorial notes]]

[[Media:06-240-classnotes_for_november18th.pdf|Nov16 Lecture_notes]]

[[Media:06-240-tutorial for november16th.pdf|Nov16 tutotial_notes]]

06-240/Classnotes For Thursday November 16

2006-12-04T08:44:42Z

Gokmen:

{{06-240/Navigation}}

[[Media:06-240-nov16lec-1.jpeg|Nov16 Lecture notes: 1 of 2]]

[[Media:06-240-nov16lec-2.jpeg|Nov16 Lecture notes: 2 of 2]]

[[Media:06-240-nov16tut-1.jpeg|Nov16 Tutorial notes: 1 of 4]]

[[Media:06-240-nov16tut-2.jpeg|Nov16 Tutorial notes: 2 of 4]]

[[Media:06-240-nov16tut-3.jpeg|Nov16 Tutorial notes: 3 of 4]]

[[Media:06-240-nov16tut-4.jpeg|Nov16 Tutorial notes: 4 of 4]]

[[Media:MAT Lect017.pdf|Scan of Week 10 Lecture 2 notes]]

[[Media:MAT Tut010.pdf|Scan of Week 10 Tutorial notes]]

[[Media:06-240-classnotes_for_november18th.pdf|Nov16 Lecturenotes]]

[[Media:06-240-tutorial for november16th.pdf|Nov16 tutotialnotes]]

06-240/Classnotes For Thursday November 16

2006-12-04T08:44:12Z

Gokmen:

{{06-240/Navigation}}

[[Media:06-240-nov16lec-1.jpeg|Nov16 Lecture notes: 1 of 2]]

[[Media:06-240-nov16lec-2.jpeg|Nov16 Lecture notes: 2 of 2]]

[[Media:06-240-nov16tut-1.jpeg|Nov16 Tutorial notes: 1 of 4]]

[[Media:06-240-nov16tut-2.jpeg|Nov16 Tutorial notes: 2 of 4]]

[[Media:06-240-nov16tut-3.jpeg|Nov16 Tutorial notes: 3 of 4]]

[[Media:06-240-nov16tut-4.jpeg|Nov16 Tutorial notes: 4 of 4]]

[[Media:MAT Lect017.pdf|Scan of Week 10 Lecture 2 notes]]

[[Media:MAT Tut010.pdf|Scan of Week 10 Tutorial notes]]

[[Media:06-240-classnotes_for_november18th.pdf|Nov16 Lecture notes]]

[[Media:06-240-tutorial for november16th.pdf|Nov16 tutotialnotes]]

06-240/Classnotes For Thursday November 16

2006-12-04T08:43:38Z

Gokmen:

{{06-240/Navigation}}

[[Media:06-240-nov16lec-1.jpeg|Nov16 Lecture notes: 1 of 2]]

[[Media:06-240-nov16lec-2.jpeg|Nov16 Lecture notes: 2 of 2]]

[[Media:06-240-nov16tut-1.jpeg|Nov16 Tutorial notes: 1 of 4]]

[[Media:06-240-nov16tut-2.jpeg|Nov16 Tutorial notes: 2 of 4]]

[[Media:06-240-nov16tut-3.jpeg|Nov16 Tutorial notes: 3 of 4]]

[[Media:06-240-nov16tut-4.jpeg|Nov16 Tutorial notes: 4 of 4]]

[[Media:MAT Lect017.pdf|Scan of Week 10 Lecture 2 notes]]

[[Media:MAT Tut010.pdf|Scan of Week 10 Tutorial notes]]

[[Media:06-240-classnotes_for_november18th.pdf|Nov16 Lecture notes: 1 of 2]]

[[Media:06-240-tutorial for november16th.pdf|Nov16 tutotialnotes: 1 of 2]]

File:06-240-tutorial for november16th.pdf

2006-12-04T08:42:18Z

Gokmen:

06-240/Classnotes For Thursday November 16

2006-12-04T08:24:30Z

Gokmen:

File:06-240-classnotes for november18th.pdf

2006-12-04T08:21:30Z

Gokmen:

06-240/Classnotes For Tuesday November 14

2006-11-18T02:51:05Z

Gokmen:

{{06-240/Navigation}}

[[Media:06-240-nov14lec-1.jpeg|Nov14 Lecture notes: 1 of 4]]

[[Media:06-240-nov14lec-2.jpeg|Nov14 Lecture notes: 2 of 4]]

[[Media:06-240-nov14lec-3.jpeg|Nov14 Lecture notes: 3 of 4]]

[[Media:06-240-nov14lec-4.jpeg|Nov14 Lecture notes: 4 of 4]]

[[Media:06-240-november14th-lecture.pdf|November14th-Lecture notes]]

File:06-240-november14th-lecture.pdf

2006-11-18T02:47:42Z

Gokmen:

File:06-240-02-Nov.tut.pdf

2006-11-03T05:54:34Z

Gokmen:

06-240/Classnotes For Thursday November 2

2006-11-03T05:15:15Z

Gokmen:

{{06-240/Navigation}}

[[Media:06-240-nov2lec.pdf|Nov02 Lecture notes]]

[[Media:06-240-02-November.pdf|November2nd-Lecture notes]]

File:06-240-02-November.pdf

2006-11-03T05:12:32Z

Gokmen:

06-240/Classnotes For Tuesday October 31

2006-11-03T05:00:30Z

Gokmen:

[[Media:06-240-31.Nov.06.pdf|Oct31-Lecture notes]]

File:06-240-31.Nov.06.pdf

2006-11-03T04:54:12Z

Gokmen:

06-240/Classnotes For Thursday October 26

2006-10-27T18:56:59Z

Gokmen:

[[Media:MAT Lect011.pdf|Scan of Week 7 Lecture notes]]

[[Media:MAT Tut006.pdf|Scan of Week 7 Tutorial notes]]

[[Media:06-240-lecture-october26.pdf|oct26 Lecture notes]]

[[Media:06-240-tutorial-october26.pdf|oct26 Tutorial notes]]

File:06-240-lecture-october26.pdf

2006-10-27T18:51:24Z

Gokmen:

[[Media:06-240-lecture-october26.pdf|oct26 Lecture notes]]

File:06-240-tutorial-october26.pdf

2006-10-27T18:38:58Z

Gokmen:

File:06-240-lecture-october26.pdf

2006-10-27T18:36:02Z

Gokmen:

File:06-240-lecture-october17.pdf

2006-10-18T20:56:25Z

Gokmen:

File:06-240-tutorial-12-october.pdf

2006-10-18T20:44:00Z

Gokmen:

File:06-240-lecture-12-october.pdf

2006-10-18T20:28:16Z

Gokmen:

File:06-240-lecture-10-october.pdf

2006-10-18T20:20:34Z

Gokmen:

File:06-240-lecture-05-october.pdf

2006-10-18T20:03:54Z

Gokmen:

File:06-240-Lecture-03-October.pdf

2006-10-18T19:51:45Z

Gokmen:

File:06-240-tutorial-28-september.pdf

2006-10-18T19:31:56Z

Gokmen:

File:06-240-Lecture-28-september.pdf

2006-10-18T19:12:54Z

Gokmen:

File:06-240-Lecture-26-september.pdf

2006-10-18T19:04:50Z

Gokmen:

File:06-240-tutorial-21-september.pdf

2006-10-18T18:49:08Z

Gokmen:

File:06-240-Lecture-21-september.pdf

2006-10-18T18:36:53Z

Gokmen:

File:06-240-Lecture-19-september.pdf

2006-10-18T18:28:57Z

Gokmen:

File:06-240-14-sept-tutorial.pdf

2006-10-18T18:14:43Z

Gokmen:

File:06-240-14-September.pdf

2006-10-18T17:59:47Z

Gokmen: