User:IssaRice/Linear algebra/Linear transformation vs matrix views - Revision history

https://machinelearning.subwiki.org/w/index.php?action=history&feed=atom&title=User%3AIssaRice%2FLinear_algebra%2FLinear_transformation_vs_matrix_views User:IssaRice/Linear algebra/Linear transformation vs matrix views - Revision history 2026-06-02T09:03:23Z Revision history for this page on the wiki MediaWiki 1.41.2 https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Linear_transformation_vs_matrix_views&diff=2306&oldid=prev IssaRice at 03:26, 31 July 2019 2019-07-31T03:26:12Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:26, 31 July 2019</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l44">Line 44:</td> <td colspan="2" class="diff-lineno">Line 44:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>given a property P of matrices, we can say that P is preserved under change of coordinates iff <math>P(A) \iff P(QAR^{-1})</math> for every invertible matrices <math>Q,R</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>given a property P of matrices, we can say that P is preserved under change of coordinates iff <math>P(A) \iff P(QAR^{-1})</math> for every invertible matrices <math>Q,R</math></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">this question discusses some of these: https://math.stackexchange.com/questions/3033378/what-properties-of-a-linear-map-can-be-determined-from-its-matrix</ins></div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Linear_transformation_vs_matrix_views&diff=2305&oldid=prev IssaRice at 02:55, 31 July 2019 2019-07-31T02:55:56Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:55, 31 July 2019</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l27">Line 27:</td> <td colspan="2" class="diff-lineno">Line 27:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(1) if <math>T</math> is injective and <math>\beta,\gamma</math> are any bases, then <math>A := [T]_\beta^\gamma</math> is injective</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(1) if <math>T</math> is injective and <math>\beta,\gamma</math> are any bases, then <math>A := [T]_\beta^\gamma</math> is injective</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>(2) if <math>A</math> is injective and <math>\beta,\gamma</math> <del style="font-weight: bold; text-decoration: none;">are any bases </del>are any bases, then for any <math>T</math> such that <math>A = [T]_\beta^\gamma</math> we have that <math>T</math> is injective</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>(2) if <math>A</math> is injective and <math>\beta,\gamma</math> are any bases, then for any <math>T</math> such that <math>A = [T]_\beta^\gamma</math> we have that <math>T</math> is injective</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>I think these can be combined into:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>I think these can be combined into:</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Linear_transformation_vs_matrix_views&diff=2304&oldid=prev IssaRice at 01:43, 31 July 2019 2019-07-31T01:43:36Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:43, 31 July 2019</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l43">Line 43:</td> <td colspan="2" class="diff-lineno">Line 43:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In particular, if we're working in <math>\mathbf R^n</math>, then the standard basis is orthonormal, so <math>T</math> is normal iff <math>A</math> is normal, where <math>A</math> is the matrix of <math>T</math> with respect to the standard basis. This is why you almost never see the distinction between "normal linear map" and "normal matrix". but as soon as you start using some other basis that isn't orthonormal, the distinction becomes important. Another way to say this is that when we say a matrix is normal, we're automatically assuming we're using the standard basis (or at least an orthonormal basis), because it is possible to adversarially select a basis under which the corresponding linear map is not normal.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In particular, if we're working in <math>\mathbf R^n</math>, then the standard basis is orthonormal, so <math>T</math> is normal iff <math>A</math> is normal, where <math>A</math> is the matrix of <math>T</math> with respect to the standard basis. This is why you almost never see the distinction between "normal linear map" and "normal matrix". but as soon as you start using some other basis that isn't orthonormal, the distinction becomes important. Another way to say this is that when we say a matrix is normal, we're automatically assuming we're using the standard basis (or at least an orthonormal basis), because it is possible to adversarially select a basis under which the corresponding linear map is not normal.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>given a property P of matrices, we can say that P is preserved under change of coordinates iff <math>P(A) \iff P(<del style="font-weight: bold; text-decoration: none;">QAQ</del>^{-1})</math> for every invertible <del style="font-weight: bold; text-decoration: none;">matrix </del><math>Q</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>given a property P of matrices, we can say that P is preserved under change of coordinates iff <math>P(A) \iff P(<ins style="font-weight: bold; text-decoration: none;">QAR</ins>^{-1})</math> for every invertible <ins style="font-weight: bold; text-decoration: none;">matrices </ins><math>Q<ins style="font-weight: bold; text-decoration: none;">,R</ins></math></div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Linear_transformation_vs_matrix_views&diff=2303&oldid=prev IssaRice at 01:39, 31 July 2019 2019-07-31T01:39:27Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:39, 31 July 2019</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If you've gone through linear algebra a couple of times, once via the matrix-based way and once via the linear maps-based way, then you should know that certain adjectives are applied to both matrices and to linear maps. For instance, we might talk about an injective matrix and also talk about an injective linear map. But it is almost never discussed (to my knowledge) whether these definitions actually correspond to each other in any way, or what the precise correspondence is.</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Given an <math>m\times n</math> matrix <math>A</math> we can define a linear map <math>T : \mathbf R^n \to \mathbf R^m</math> by <math>T(x) = Ax</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Given an <math>m\times n</math> matrix <math>A</math> we can define a linear map <math>T : \mathbf R^n \to \mathbf R^m</math> by <math>T(x) = Ax</math>.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Linear_transformation_vs_matrix_views&diff=2302&oldid=prev IssaRice at 01:32, 31 July 2019 2019-07-31T01:32:48Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:32, 31 July 2019</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l39">Line 39:</td> <td colspan="2" class="diff-lineno">Line 39:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for all <math>T,A</math> and all orthonormal <math>\beta,\gamma</math> such that <math>A = [T]_\beta^\gamma</math>: T normal iff A normal.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for all <math>T,A</math> and all orthonormal <math>\beta,\gamma</math> such that <math>A = [T]_\beta^\gamma</math>: T normal iff A normal.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In particular, if we're working in <math>\mathbf R^n</math>, then the standard basis is orthonormal, so <math>T</math> is normal iff <math>A</math> is normal, where <math>A</math> is the matrix of <math>T</math> with respect to the standard basis. This is why you almost never see the distinction between "normal linear map" and "normal matrix". but as soon as you start using some other basis that isn't orthonormal, the distinction becomes important.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In particular, if we're working in <math>\mathbf R^n</math>, then the standard basis is orthonormal, so <math>T</math> is normal iff <math>A</math> is normal, where <math>A</math> is the matrix of <math>T</math> with respect to the standard basis. This is why you almost never see the distinction between "normal linear map" and "normal matrix". but as soon as you start using some other basis that isn't orthonormal, the distinction becomes important<ins style="font-weight: bold; text-decoration: none;">. Another way to say this is that when we say a matrix is normal, we're automatically assuming we're using the standard basis (or at least an orthonormal basis), because it is possible to adversarially select a basis under which the corresponding linear map is not normal</ins>.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>given a property P of matrices, we can say that P is preserved under change of coordinates iff <math>P(A) \iff P(QAQ^{-1})</math> for every invertible matrix <math>Q</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>given a property P of matrices, we can say that P is preserved under change of coordinates iff <math>P(A) \iff P(QAQ^{-1})</math> for every invertible matrix <math>Q</math></div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Linear_transformation_vs_matrix_views&diff=2301&oldid=prev IssaRice at 01:29, 31 July 2019 2019-07-31T01:29:37Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:29, 31 July 2019</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l40">Line 40:</td> <td colspan="2" class="diff-lineno">Line 40:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In particular, if we're working in <math>\mathbf R^n</math>, then the standard basis is orthonormal, so <math>T</math> is normal iff <math>A</math> is normal, where <math>A</math> is the matrix of <math>T</math> with respect to the standard basis. This is why you almost never see the distinction between "normal linear map" and "normal matrix". but as soon as you start using some other basis that isn't orthonormal, the distinction becomes important.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In particular, if we're working in <math>\mathbf R^n</math>, then the standard basis is orthonormal, so <math>T</math> is normal iff <math>A</math> is normal, where <math>A</math> is the matrix of <math>T</math> with respect to the standard basis. This is why you almost never see the distinction between "normal linear map" and "normal matrix". but as soon as you start using some other basis that isn't orthonormal, the distinction becomes important.</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">given a property P of matrices, we can say that P is preserved under change of coordinates iff <math>P(A) \iff P(QAQ^{-1})</math> for every invertible matrix <math>Q</math></ins></div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Linear_transformation_vs_matrix_views&diff=2300&oldid=prev IssaRice at 01:25, 31 July 2019 2019-07-31T01:25:12Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:25, 31 July 2019</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l38">Line 38:</td> <td colspan="2" class="diff-lineno">Line 38:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for all <math>T,A</math> and all orthonormal <math>\beta,\gamma</math> such that <math>A = [T]_\beta^\gamma</math>: T normal iff A normal.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for all <math>T,A</math> and all orthonormal <math>\beta,\gamma</math> such that <math>A = [T]_\beta^\gamma</math>: T normal iff A normal.</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In particular, if we're working in <math>\mathbf R^n</math>, then the standard basis is orthonormal, so <math>T</math> is normal iff <math>A</math> is normal, where <math>A</math> is the matrix of <math>T</math> with respect to the standard basis. This is why you almost never see the distinction between "normal linear map" and "normal matrix". but as soon as you start using some other basis that isn't orthonormal, the distinction becomes important.</ins></div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Linear_transformation_vs_matrix_views&diff=2299&oldid=prev IssaRice at 01:21, 31 July 2019 2019-07-31T01:21:06Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:21, 31 July 2019</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l34">Line 34:</td> <td colspan="2" class="diff-lineno">Line 34:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>What about properties like "normal", "self-adjoint", "isometry"? what about trace, determinant? sum of the first column?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>What about properties like "normal", "self-adjoint", "isometry"? what about trace, determinant? sum of the first column?</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">It seems like for "normal", we don't have this for every choice of bases. I think the correct statement is:</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">for all <math>T,A</math> and all orthonormal <math>\beta,\gamma</math> such that <math>A = [T]_\beta^\gamma</math>: T normal iff A normal.</ins></div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Linear_transformation_vs_matrix_views&diff=2298&oldid=prev IssaRice at 01:18, 31 July 2019 2019-07-31T01:18:38Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:18, 31 July 2019</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l32">Line 32:</td> <td colspan="2" class="diff-lineno">Line 32:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Potential proof strategy that might not be so tedious: basically imagine that A is always written in the standard basis. Then consider <math>[T]_\sigma^{\sigma'}</math> where <math>\sigma,\sigma'</math> are the standard bases in R^n and R^m. Then <math>A = [I]_\gamma^{\sigma'}[T]_\beta^\gamma[I]_\sigma^\beta</math>. Then somehow we can use the fact that change of coordinate matrices are invertible.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Potential proof strategy that might not be so tedious: basically imagine that A is always written in the standard basis. Then consider <math>[T]_\sigma^{\sigma'}</math> where <math>\sigma,\sigma'</math> are the standard bases in R^n and R^m. Then <math>A = [I]_\gamma^{\sigma'}[T]_\beta^\gamma[I]_\sigma^\beta</math>. Then somehow we can use the fact that change of coordinate matrices are invertible.</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">What about properties like "normal", "self-adjoint", "isometry"? what about trace, determinant? sum of the first column?</ins></div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Linear_transformation_vs_matrix_views&diff=2151&oldid=prev IssaRice at 22:27, 27 June 2019 2019-06-27T22:27:39Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 22:27, 27 June 2019</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l31">Line 31:</td> <td colspan="2" class="diff-lineno">Line 31:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(3) for all <math>T,A,\beta,\gamma</math> such that <math>A = [T]_\beta^\gamma</math>: T injective iff A injective.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(3) for all <math>T,A,\beta,\gamma</math> such that <math>A = [T]_\beta^\gamma</math>: T injective iff A injective.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Potential proof strategy that might not be so tedious: basically imagine that A is always written in the standard basis. Then consider <math>[T]_\sigma^{\sigma'}</math> where <math>\sigma,\sigma'</math> are the standard bases in R^n and R^m. Then <math>A = [I]_\gamma^{\sigma'}[T]_\beta^\gamma[I]_\sigma^\beta</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Potential proof strategy that might not be so tedious: basically imagine that A is always written in the standard basis. Then consider <math>[T]_\sigma^{\sigma'}</math> where <math>\sigma,\sigma'</math> are the standard bases in R^n and R^m. Then <math>A = [I]_\gamma^{\sigma'}[T]_\beta^\gamma[I]_\sigma^\beta</math><ins style="font-weight: bold; text-decoration: none;">. Then somehow we can use the fact that change of coordinate matrices are invertible</ins>.</div></td></tr> </table>

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