14-240/Tutorial-October14 - Revision history

Boyang.wu: /* Nikita */

2014-12-08T19:08:52Z

Nikita

← Older revision		Revision as of 15:08, 8 December 2014
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	==Nikita==		==Nikita==
	[[File:1014.240.pdf]]		[[File:1014.240.pdf]]

			==Scanned Tutorial Notes by [[User Boyang.wu\|Boyang.wu]]==
			[[File:Tut6.pdf]]

Bug: /* Problem 5h) in Homework 3 for all Fields */

2014-11-29T15:42:49Z

Problem 5h) in Homework 3 for all Fields

← Older revision		Revision as of 11:42, 29 November 2014
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	Show both (a) and (b) to be on the safe side.		Show both (a) and (b) to be on the safe side.

	====Problem 5h) in Homework 3 for all Fields====		====Problem 5h) on Page 34 in Homework 3 for all Fields====


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	\end{pmatrix}		\end{pmatrix}
	\in span(S) \iff char(F)=2</math>. ''Q.E.D.''		\in span(S) \iff char(F)=2</math>. ''Q.E.D.''



	====A Field Problem====		====A Field Problem====

Bug: /* Elementary and (Not So Elementary) Errors in Homework */

2014-11-23T01:53:31Z

Elementary and (Not So Elementary) Errors in Homework

← Older revision		Revision as of 21:53, 22 November 2014
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	(1) ~~'''~~Bad Notation'''		'''(1) Bad Notation'''


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	(2) ~~'''~~Algorithm vs. Proof (Boris's Section Only)'''		'''(2) Algorithm vs. Proof (Boris's Section Only)'''

	When solving a problem that requires a solution to a linear equation, it is not always obvious which of the following you should show:		When solving a problem that requires a solution to a linear equation, it is not always obvious which of the following you should show:

Bug: /* A Dimension Problem */

2014-11-23T01:39:30Z

A Dimension Problem

← Older revision		Revision as of 21:39, 22 November 2014
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	::::Since <math>B_2 \subset (B_1 \cup B_2)</math>, then <math>span(B_2) = W_2 \subset span(B_1 \cup B_2)</math>.		::::Since <math>B_2 \subset (B_1 \cup B_2)</math>, then <math>span(B_2) = W_2 \subset span(B_1 \cup B_2)</math>.

	::::Then <math>(W_1 \cup W_2) \subset span(B_1 \cup B_2)</math>.		::::Then <math>(W_1 \cup W_2) \subset span(B_1 \cup B_2)</math> and <math>span(W_1 \cup W_2) \subset span(B_1 \cup B_2)</math>.

	:::Then <math>span(B_1 \cup B_2) = span(W_1 \cup W_2)</math>.		:::Then <math>span(B_1 \cup B_2) = span(W_1 \cup W_2)</math>.

Bug at 01:31, 23 November 2014

2014-11-23T01:31:26Z

← Older revision		Revision as of 21:31, 22 November 2014
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	Firstly, <math>span(M_1, M_2, M_2)</math> is the set of all linear combinations of <math>M_1, M_2, M_3</math>. To equate it to a single		Firstly, <math>span(M_1, M_2, M_2)</math> is the set of all linear combinations of <math>M_1, M_2, M_3</math>. To equate it to a single symmetric <math>2 \times 2</math> matrix makes no sense. Secondly, the elements <math>a, b, c, d</math> are undefined. What are they suppose to represent? Rational numbers? Real numbers? Members of the field of two elements? The following way of writing erases those issues:

	symmetric
	<math>2 \times 2</math> matrix makes no sense. Secondly, the elements <math>a, b, c, d</math> are undefined. What are they suppose to

	represent? Rational numbers? Real numbers? Members of the field of two elements? The following way of writing erases

	those issues:


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	(2) '''Algorithm vs. Proof (Boris's Section Only)'''		(2) '''Algorithm vs. Proof (Boris's Section Only)'''

	When solving a problem that requires a solution to a linear equation, it is not always obvious which of the following you		When solving a problem that requires a solution to a linear equation, it is not always obvious which of the following you should show:

	should show:

	:a) An algorithm for finding the solution		:a) An algorithm for finding the solution

	:b) A proof that a solution is correct		:b) A proof that a solution is correct

	If the problem asks to solve a linear equation, then just show (a). Otherwise, consider problems such as this:		If the problem asks to solve a linear equation, then just show (a). Otherwise, consider problems such as this:

Bug: /* Elementary and (Not So Elementary) Errors in Homework */

2014-10-30T02:00:14Z

Elementary and (Not So Elementary) Errors in Homework

← Older revision		Revision as of 22:00, 29 October 2014
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	<math>2 \times 2</math> matrix makes no sense. Secondly, the elements <math>a, b, c, d</math> are undefined. What are they suppose to		<math>2 \times 2</math> matrix makes no sense. Secondly, the elements <math>a, b, c, d</math> are undefined. What are they suppose to

	represent? Rational numbers? Real numbers? Members of the field of two elements? The following way of writing erases ~~those~~		represent? Rational numbers? Real numbers? Members of the field of two elements? The following way of writing erases

	issues:		those issues:

Bug: /* Elementary and (Not So Elementary) Errors in Homework */

2014-10-30T01:59:49Z

Elementary and (Not So Elementary) Errors in Homework

← Older revision		Revision as of 21:59, 29 October 2014
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	<math>2 \times 2</math> matrix makes no sense. Secondly, the elements <math>a, b, c, d</math> are undefined. What are they suppose to		<math>2 \times 2</math> matrix makes no sense. Secondly, the elements <math>a, b, c, d</math> are undefined. What are they suppose to

	represent? Rational numbers? Real numbers? Members of the field of two elements? The following way of writing erases those ~~issues:~~		represent? Rational numbers? Real numbers? Members of the field of two elements? The following way of writing erases those

			issues:

Bug: /* Elementary and (Not So Elementary) Errors in Homework */

2014-10-30T01:57:30Z

Elementary and (Not So Elementary) Errors in Homework

← Older revision		Revision as of 21:57, 29 October 2014
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	::::::<math>		:::::<math>
	span(M_1, M_2, M_3) = \{		span(M_1, M_2, M_3) = \{
	\begin{pmatrix}		\begin{pmatrix}

Bug: /* Elementary and (Not So Elementary) Errors in Homework */

2014-10-30T01:57:06Z

Elementary and (Not So Elementary) Errors in Homework

← Older revision		Revision as of 21:57, 29 October 2014
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	::::::::<math>		::::::<math>
	span(M_1, M_2, M_3) = \{		span(M_1, M_2, M_3) = \{
	\begin{pmatrix}		\begin{pmatrix}

Bug: /* Elementary and (Not So Elementary) Errors in Homework */

2014-10-30T01:56:01Z

Elementary and (Not So Elementary) Errors in Homework

← Older revision		Revision as of 21:56, 29 October 2014
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			Consider these three matrices:
	Let
	::::::::<math>		::::::::<math>
	M_1 =		M_1 =
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	~~be matrices.~~ We want to equate <math>span(M_1, M_2, M_3)</math> to the set of all symmetric <math>2 \times 2</math> matrices. Here is the wrong way to write this:		We want to equate <math>span(M_1, M_2, M_3)</math> to the set of all symmetric <math>2 \times 2</math> matrices. Here is the wrong way to write this:


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	\end{pmatrix}		\end{pmatrix}
	:a, b, c \in F \}		:a, b, c \in F \}
		⚫	</math> where <math>F</math> is an arbitrary field.
	</math>

⚫	where <math>F</math> is an arbitrary field.