06-240/Classnotes For Thursday November 9 - Revision history

Drorbn: Reverted edit of 76.98.229.83, changed back to last version by Alla

2007-05-28T14:50:20Z

Reverted edit of 76.98.229.83, changed back to last version by Alla

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 |-
 |'''Problem.''' Find the rank (the dimension of the image) of a linear transformation <math>T</math> whose matrix representation is the matrix A shown on the right.
-|<math>A=\begin{pmatrix}0</math>
+|<math>A=\begin{pmatrix}0&2&4&2&2\\4&4&4&8&0\\8&2&0&10&2\\6&3&2&9&1\end{pmatrix}</math>.

2007-05-28T06:37:20Z

@@ Line 5: / Line 5: @@
 |-
 |'''Problem.''' Find the rank (the dimension of the image) of a linear transformation <math>T</math> whose matrix representation is the matrix A shown on the right.
-|<math>A=\begin{pmatrix}0&2&4&2&2\\4&4&4&8&0\\8&2&0&10&2\\6&3&2&9&1\end{pmatrix}</math>.
+|<math>A=\begin{pmatrix}0</math>

2006-11-26T17:23:05Z

formatted

← Older revision		Revision as of 13:23, 26 November 2006
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	[[Media:Lect015.pdf\|Scan of Week 9 Lecture 2 notes]]		[[Media:Lect015.pdf\|Scan of Week 9 Lecture 2 notes]]

⚫	[[Media:Tut009.pdf\|Scan of Week 9 Tutorial notes]]

	==Tutorial Notes==		==Tutorial Notes==
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	[[Media:06-240-nov09tut-3.jpeg\|Nov09 Lecture notes 3 of 3]]		[[Media:06-240-nov09tut-3.jpeg\|Nov09 Lecture notes 3 of 3]]

		⚫	[[Media:Tut009.pdf\|Scan of Week 9 Tutorial notes]]

2006-11-16T00:00:09Z

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 {{06-240/Navigation}}
 ==Review of Last Class==
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 But the matrix we now have represents a linear transformation <math>S</math> satisfying <math>S(v_1,\,v_2,\,v_3,\,v_4\,v_5)=(w_1,\,w_2,\,w_3,\,0,\,0)</math> for some bases <math>(v_i)_{i=1}^5</math> of <math>V</math> and <math>(w_j)_{j=1}^4</math> of <math>W</math>. Thus the image (range) of <math>S</math> is spanned by <math>\{w_1,w_2,w_3\}</math>, and as these are independent, they form a basis of the image. Thus the rank of <math>S</math> is <math>3</math>. Going backward through the "matrix reduction" process above and repeatedly using theorems 1 and 2, we find that the rank of <math>T</math> must also be <math>3</math>.

2006-11-15T03:50:50Z

Uploaded Week 9 Lecture 2 notes and Tutorial notes

← Older revision		Revision as of 23:50, 14 November 2006
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	[[Media:06-240-nov09tut-3.jpeg\|Nov09 Lecture notes 3 of 3]]		[[Media:06-240-nov09tut-3.jpeg\|Nov09 Lecture notes 3 of 3]]

			[[Media:Lect015.pdf\|Scan of Week 9 Lecture 2 notes]]

			[[Media:Tut009.pdf\|Scan of Week 9 Tutorial notes]]

2006-11-15T02:51:44Z

← Older revision		Revision as of 22:51, 14 November 2006
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	{{06-240/Navigation}}		{{06-240/Navigation}}

			==Tutorial Notes==

			[[Media:06-240-nov09tut-1.jpeg\|Nov09 Lecture notes 1 of 3]]


			[[Media:06-240-nov09tut-2.jpeg\|Nov09 Lecture notes 2 of 3]]


			[[Media:06-240-nov09tut-3.jpeg\|Nov09 Lecture notes 3 of 3]]


	==Review of Last Class==		==Review of Last Class==

2006-11-10T04:40:07Z

'Multiply a row/column by a nonzero scalar'

← Older revision		Revision as of 00:40, 10 November 2006
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	\|'''Theorem 2.''' The following row/column operations can be applied to a matrix <math>A</math> by multiplying it on the left/right (respectively) by certain ''invertible'' "elementary matrices":		\|'''Theorem 2.''' The following row/column operations can be applied to a matrix <math>A</math> by multiplying it on the left/right (respectively) by certain ''invertible'' "elementary matrices":
	# Swap two rows/columns		# Swap two rows/columns
	# Multiply a row/column by a scalar.		# Multiply a row/column by a nonzero scalar.
	# Add a multiple of one row/column to another row/column.		# Add a multiple of one row/column to another row/column.
	\|		\|

2006-11-09T04:52:35Z

Review of Last Class

← Older revision		Revision as of 00:52, 9 November 2006
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	==Review of Last Class==		==Review of Last Class==
	[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0001.jpg]]
	[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0002.jpg]]
	[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0003.jpg]]
	[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0004.jpg]]

	{\|		{\|
	\|-		\|-

2006-11-09T04:43:58Z

Review of Last Class

← Older revision		Revision as of 00:43, 9 November 2006
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	==Review of Last Class==		==Review of Last Class==
			[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0001.jpg]]
			[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0002.jpg]]
			[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0003.jpg]]
			[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0004.jpg]]

	{\|		{\|

2006-11-08T16:00:14Z

← Older revision		Revision as of 12:00, 8 November 2006
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	\|'''Theorem 1.''' If <math>T:V\to W</math> is a linear transformation and <math>P:V\to V</math> and <math>Q:W\to W</math> are ''invertible'' linear transformations, then the rank of <math>T</math> is the same as the rank of <math>QTP</math>.		\|'''Theorem 1.''' If <math>T:V\to W</math> is a linear transformation and <math>P:V\to V</math> and <math>Q:W\to W</math> are ''invertible'' linear transformations, then the rank of <math>T</math> is the same as the rank of <math>QTP</math>.
	\|		\|
	\|'''Proof.'''~~ ~~Owed.		\|'''Proof.''' Owed.
	\|- valign=bottom		\|- valign=bottom
	\|'''Theorem 2.''' The following row/column operations can be applied to a matrix <math>A</math> by multiplying it on the left/right (respectively) by certain ''invertible'' "elementary matrices":		\|'''Theorem 2.''' The following row/column operations can be applied to a matrix <math>A</math> by multiplying it on the left/right (respectively) by certain ''invertible'' "elementary matrices":
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	# Add a multiple of one row/column to another row/column.		# Add a multiple of one row/column to another row/column.
	\|		\|
	\|'''Proof.'''~~ ~~Semi-owed.		\|'''Proof.''' Semi-owed.
	\|}		\|}