Pub Date : 2018-08-09DOI: 10.2174/9781681087115118010007
{"title":"Nonlinear Transformations","authors":"","doi":"10.2174/9781681087115118010007","DOIUrl":"https://doi.org/10.2174/9781681087115118010007","url":null,"abstract":"","PeriodicalId":286524,"journal":{"name":"Transformations: A Mathematical Approach- Fundamental Concepts","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123654284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-09DOI: 10.2174/9781681087115118010012
{"title":"Cybernetics","authors":"","doi":"10.2174/9781681087115118010012","DOIUrl":"https://doi.org/10.2174/9781681087115118010012","url":null,"abstract":"","PeriodicalId":286524,"journal":{"name":"Transformations: A Mathematical Approach- Fundamental Concepts","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132421368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-09DOI: 10.2174/9781681087115118010006
Last week, you saw that we can think of 2 × 2 matrices as functions of matrices as functions the Cartesian plane—that is, matrix multiplication represents some sort of two-dimensional transformation. The obvious question is: what sorts of two-dimensional transformations can we represent with matrices in this way? Another important question is, given some matrix, how can we tell what transformation it represents? Last week, you looked at a few examples— but how do we know we can tell what transformation a matrix represents just by looking at its action on a few points? For example, one of the matrices you looked transformed four points making up the vertices of a square into a slightly rotated square—but how do we know this matrix would have the same effect on other points? Perhaps it only rotates points near the origin, and leaves some farther-away points alone, or maybe it rotates some points clockwise and some points counterclockwise, or maybe.. . there are endless possibilities. It turns out that we can actually tell what transformation a matrix repre-matrix functions are simple sents by looking at only a few examples, and that there are really only a few fundamental sorts of transformations that can be represented by matrices (fortunately, they're very useful ones!). The fundamental result we will show is this: If we know what a matrix does to the special points (1, 0) and (0, 1), then we know everything there is to know about it! Intuitively, this says that transformations represented by matrices can't do anything very " strange ". They are so orderly and regular that we can know everything there is to know about them by just looking at what they do to two specific points. Let's see why this is true!
{"title":"Linear Transformations","authors":"","doi":"10.2174/9781681087115118010006","DOIUrl":"https://doi.org/10.2174/9781681087115118010006","url":null,"abstract":"Last week, you saw that we can think of 2 × 2 matrices as functions of matrices as functions the Cartesian plane—that is, matrix multiplication represents some sort of two-dimensional transformation. The obvious question is: what sorts of two-dimensional transformations can we represent with matrices in this way? Another important question is, given some matrix, how can we tell what transformation it represents? Last week, you looked at a few examples— but how do we know we can tell what transformation a matrix represents just by looking at its action on a few points? For example, one of the matrices you looked transformed four points making up the vertices of a square into a slightly rotated square—but how do we know this matrix would have the same effect on other points? Perhaps it only rotates points near the origin, and leaves some farther-away points alone, or maybe it rotates some points clockwise and some points counterclockwise, or maybe.. . there are endless possibilities. It turns out that we can actually tell what transformation a matrix repre-matrix functions are simple sents by looking at only a few examples, and that there are really only a few fundamental sorts of transformations that can be represented by matrices (fortunately, they're very useful ones!). The fundamental result we will show is this: If we know what a matrix does to the special points (1, 0) and (0, 1), then we know everything there is to know about it! Intuitively, this says that transformations represented by matrices can't do anything very \" strange \". They are so orderly and regular that we can know everything there is to know about them by just looking at what they do to two specific points. Let's see why this is true!","PeriodicalId":286524,"journal":{"name":"Transformations: A Mathematical Approach- Fundamental Concepts","volume":"548 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127660394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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