User:IssaRice/Linear algebra/Geometry of linear transformations - Revision history

IssaRice at 01:31, 29 May 2020

2020-05-29T01:31:42Z

← Older revision		Revision as of 01:31, 29 May 2020
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	this video https://youtu.be/dtvM-CzNe50?t=229 lists the following: (1) keeps gridlines parallel, (2) keep gridlines evenly spaced, (3) keeps origin stationary.		this video https://youtu.be/dtvM-CzNe50?t=229 lists the following: (1) keeps gridlines parallel, (2) keep gridlines evenly spaced, (3) keeps origin stationary.

			https://math.stackexchange.com/questions/2045859/relationship-between-properties-of-linear-transformations-algebraically-and-visu?noredirect=1&lq=1

IssaRice at 01:21, 30 April 2020

2020-04-30T01:21:52Z

← Older revision		Revision as of 01:21, 30 April 2020
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	(the above is all just motivation. when i feel like it i'll fill in the rest)		(the above is all just motivation. when i feel like it i'll fill in the rest)

			this video https://youtu.be/dtvM-CzNe50?t=229 lists the following: (1) keeps gridlines parallel, (2) keep gridlines evenly spaced, (3) keeps origin stationary.

IssaRice at 00:58, 17 January 2020

2020-01-17T00:58:25Z

← Older revision		Revision as of 00:58, 17 January 2020
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	There are various "geometric properties" that transformations can have, such as "[https://en.wikipedia.org/wiki/Isometry preserves lengths]", "[https://en.wikipedia.org/wiki/Conformal_map preserves angles]", and so forth. We might wonder, what kind of similar property could characterize linear transformations? Come to think of it, the definition often used for linear maps, that of T(v+w)=Tv + Tw, and T(av)=aTv, is very convenient from a symbolic-manipulation point of view, but gives hardly any idea what a linear map could ''mean''.		There are various "geometric properties" that transformations can have, such as "[https://en.wikipedia.org/wiki/Isometry preserves lengths]", "[https://en.wikipedia.org/wiki/Conformal_map preserves angles]", and so forth. We might wonder, what kind of similar property could characterize linear transformations? Come to think of it, the definition often used for linear maps, that of T(v+w)=Tv + Tw, and T(av)=aTv, is very convenient from a symbolic-manipulation point of view, but gives hardly any idea what a linear map could ''mean''.

	For example, it's pretty obvious that linear maps preserve the origin. They also send lines to lines. They also send any equally-spaced-apart collinear points to equally-spaced-apart collinear points (not necessarily spaced apart at the same rate). They send lines through the origin to lines through the origin. They send parallel lines to parallel lines. "the image of the intersection of a pair of lines is the intersection of the lines’ images", "maps paralellograms to parallelograms" [https://math.stackexchange.com/a/3200938/35525]. sends circles to ellipses (or maybe better to phrase as "sends ellipses to ellipses").		For example, it's pretty obvious that linear maps preserve the origin. They also send lines to lines. They also send any equally-spaced-apart collinear points to equally-spaced-apart collinear points (not necessarily spaced apart at the same rate). They send lines through the origin to lines through the origin. They send parallel lines to parallel lines. "the image of the intersection of a pair of lines is the intersection of the lines’ images", "maps paralellograms to parallelograms" [https://math.stackexchange.com/a/3200938/35525]. sends circles to ellipses (or maybe better to phrase as "sends ellipses to ellipses"). sends squares to parallelograms? parallelograms to parallelograms?

	What's ''not'' obvious is what is the "minimal list" of properties that must be preserved, such that if a function preserves those properties, then it is necessarily linear. In other words, proving linear implies [list of properties] is easy, but proving the converse, [list of properties] implies linear, is harder. It's also hard to figure out ''which'' list of properties one ought to use.		What's ''not'' obvious is what is the "minimal list" of properties that must be preserved, such that if a function preserves those properties, then it is necessarily linear. In other words, proving linear implies [list of properties] is easy, but proving the converse, [list of properties] implies linear, is harder. It's also hard to figure out ''which'' list of properties one ought to use.

	(the above is all just motivation. when i feel like it i'll fill in the rest)		(the above is all just motivation. when i feel like it i'll fill in the rest)

IssaRice at 00:57, 17 January 2020

2020-01-17T00:57:56Z

← Older revision		Revision as of 00:57, 17 January 2020
Line 1:		Line 1:
	There are various "geometric properties" that transformations can have, such as "[https://en.wikipedia.org/wiki/Isometry preserves lengths]", "[https://en.wikipedia.org/wiki/Conformal_map preserves angles]", and so forth. We might wonder, what kind of similar property could characterize linear transformations? Come to think of it, the definition often used for linear maps, that of T(v+w)=Tv + Tw, and T(av)=aTv, is very convenient from a symbolic-manipulation point of view, but gives hardly any idea what a linear map could ''mean''.		There are various "geometric properties" that transformations can have, such as "[https://en.wikipedia.org/wiki/Isometry preserves lengths]", "[https://en.wikipedia.org/wiki/Conformal_map preserves angles]", and so forth. We might wonder, what kind of similar property could characterize linear transformations? Come to think of it, the definition often used for linear maps, that of T(v+w)=Tv + Tw, and T(av)=aTv, is very convenient from a symbolic-manipulation point of view, but gives hardly any idea what a linear map could ''mean''.

	For example, it's pretty obvious that linear maps preserve the origin. They also send lines to lines. They also send any equally-spaced-apart collinear points to equally-spaced-apart collinear points (not necessarily spaced apart at the same rate). They send lines through the origin to lines through the origin. They send parallel lines to parallel lines. "the image of the intersection of a pair of lines is the intersection of the lines’ images", "maps paralellograms to parallelograms" [https://math.stackexchange.com/a/3200938/35525].		For example, it's pretty obvious that linear maps preserve the origin. They also send lines to lines. They also send any equally-spaced-apart collinear points to equally-spaced-apart collinear points (not necessarily spaced apart at the same rate). They send lines through the origin to lines through the origin. They send parallel lines to parallel lines. "the image of the intersection of a pair of lines is the intersection of the lines’ images", "maps paralellograms to parallelograms" [https://math.stackexchange.com/a/3200938/35525]. sends circles to ellipses (or maybe better to phrase as "sends ellipses to ellipses").

	What's ''not'' obvious is what is the "minimal list" of properties that must be preserved, such that if a function preserves those properties, then it is necessarily linear. In other words, proving linear implies [list of properties] is easy, but proving the converse, [list of properties] implies linear, is harder. It's also hard to figure out ''which'' list of properties one ought to use.		What's ''not'' obvious is what is the "minimal list" of properties that must be preserved, such that if a function preserves those properties, then it is necessarily linear. In other words, proving linear implies [list of properties] is easy, but proving the converse, [list of properties] implies linear, is harder. It's also hard to figure out ''which'' list of properties one ought to use.

	(the above is all just motivation. when i feel like it i'll fill in the rest)		(the above is all just motivation. when i feel like it i'll fill in the rest)

IssaRice at 02:36, 16 January 2020

2020-01-16T02:36:49Z

← Older revision		Revision as of 02:36, 16 January 2020
Line 1:		Line 1:
	There are various "geometric properties" that transformations can have, such as "[https://en.wikipedia.org/wiki/Isometry preserves lengths]", "[https://en.wikipedia.org/wiki/Conformal_map preserves angles]", and so forth. We might wonder, what kind of similar property could characterize linear transformations? Come to think of it, the definition often used for linear maps, that of T(v+w)=Tv + Tw, and T(av)=aTv, is very convenient from a symbolic-manipulation point of view, but gives hardly any idea what a linear map could ''mean''.		There are various "geometric properties" that transformations can have, such as "[https://en.wikipedia.org/wiki/Isometry preserves lengths]", "[https://en.wikipedia.org/wiki/Conformal_map preserves angles]", and so forth. We might wonder, what kind of similar property could characterize linear transformations? Come to think of it, the definition often used for linear maps, that of T(v+w)=Tv + Tw, and T(av)=aTv, is very convenient from a symbolic-manipulation point of view, but gives hardly any idea what a linear map could ''mean''.

	For example, it's pretty obvious that linear maps preserve the origin. They also send lines to lines. They also send any equally-spaced-apart collinear points to equally-spaced-apart collinear points (not necessarily spaced apart at the same rate). They send lines through the origin to lines through the origin. They send parallel lines to parallel lines.		For example, it's pretty obvious that linear maps preserve the origin. They also send lines to lines. They also send any equally-spaced-apart collinear points to equally-spaced-apart collinear points (not necessarily spaced apart at the same rate). They send lines through the origin to lines through the origin. They send parallel lines to parallel lines. "the image of the intersection of a pair of lines is the intersection of the lines’ images", "maps paralellograms to parallelograms" [https://math.stackexchange.com/a/3200938/35525].

	What's ''not'' obvious is what is the "minimal list" of properties that must be preserved, such that if a function preserves those properties, then it is necessarily linear. In other words, proving linear implies [list of properties] is easy, but proving the converse, [list of properties] implies linear, is harder. It's also hard to figure out ''which'' list of properties one ought to use.		What's ''not'' obvious is what is the "minimal list" of properties that must be preserved, such that if a function preserves those properties, then it is necessarily linear. In other words, proving linear implies [list of properties] is easy, but proving the converse, [list of properties] implies linear, is harder. It's also hard to figure out ''which'' list of properties one ought to use.

	(the above is all just motivation. when i feel like it i'll fill in the rest)		(the above is all just motivation. when i feel like it i'll fill in the rest)

IssaRice at 07:54, 15 January 2020

2020-01-15T07:54:34Z

← Older revision		Revision as of 07:54, 15 January 2020
Line 1:		Line 1:
	There are various "geometric properties" that transformations can have, such as "[https://en.wikipedia.org/wiki/Isometry preserves lengths]", "[https://en.wikipedia.org/wiki/Conformal_map preserves angles]", and so forth. We might wonder, what kind of similar property could characterize linear transformations?		There are various "geometric properties" that transformations can have, such as "[https://en.wikipedia.org/wiki/Isometry preserves lengths]", "[https://en.wikipedia.org/wiki/Conformal_map preserves angles]", and so forth. We might wonder, what kind of similar property could characterize linear transformations? Come to think of it, the definition often used for linear maps, that of T(v+w)=Tv + Tw, and T(av)=aTv, is very convenient from a symbolic-manipulation point of view, but gives hardly any idea what a linear map could ''mean''.

	For example, it's pretty obvious that linear maps preserve the origin. They also send lines to lines. They also send any equally-spaced-apart collinear points to equally-spaced-apart collinear points (not necessarily spaced apart at the same rate). They send lines through the origin to lines through the origin. They send parallel lines to parallel lines.		For example, it's pretty obvious that linear maps preserve the origin. They also send lines to lines. They also send any equally-spaced-apart collinear points to equally-spaced-apart collinear points (not necessarily spaced apart at the same rate). They send lines through the origin to lines through the origin. They send parallel lines to parallel lines.

	What's ''not'' obvious is what is the "minimal list" of properties that must be preserved, such that if a function preserves those properties, then it is necessarily linear. In other words, proving linear implies [list of properties] is easy, but proving the converse, [list of properties] implies linear, is harder. It's also hard to figure out ''which'' list of properties one ought to use.		What's ''not'' obvious is what is the "minimal list" of properties that must be preserved, such that if a function preserves those properties, then it is necessarily linear. In other words, proving linear implies [list of properties] is easy, but proving the converse, [list of properties] implies linear, is harder. It's also hard to figure out ''which'' list of properties one ought to use.

			(the above is all just motivation. when i feel like it i'll fill in the rest)

IssaRice: Created page with "There are various "geometric properties" that transformations can have, such as "[https://en.wikipedia.org/wiki/Isometry preserves lengths]", "[https://en.wikipedia.org/wiki/C..."

2020-01-15T07:51:25Z

Created page with "There are various "geometric properties" that transformations can have, such as "[https://en.wikipedia.org/wiki/Isometry preserves lengths]", "[https://en.wikipedia.org/wiki/C..."

New page

There are various "geometric properties" that transformations can have, such as "[https://en.wikipedia.org/wiki/Isometry preserves lengths]", "[https://en.wikipedia.org/wiki/Conformal_map preserves angles]", and so forth. We might wonder, what kind of similar property could characterize linear transformations?

For example, it's pretty obvious that linear maps preserve the origin. They also send lines to lines. They also send any equally-spaced-apart collinear points to equally-spaced-apart collinear points (not necessarily spaced apart at the same rate). They send lines through the origin to lines through the origin. They send parallel lines to parallel lines.

What's ''not'' obvious is what is the "minimal list" of properties that must be preserved, such that if a function preserves those properties, then it is necessarily linear. In other words, proving linear implies [list of properties] is easy, but proving the converse, [list of properties] implies linear, is harder. It's also hard to figure out ''which'' list of properties one ought to use.