Difference between revisions of "Math 260, Spring 2012"
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* If the kernel of a linear transformation is nontrivial, does that imply that the transformation is not injective and therefore the transformation matrix is not invertible? [[User:Patrickmclaren|Patrickmclaren]] 02:11, 6 March 2012 (GMT) |
* If the kernel of a linear transformation is nontrivial, does that imply that the transformation is not injective and therefore the transformation matrix is not invertible? [[User:Patrickmclaren|Patrickmclaren]] 02:11, 6 March 2012 (GMT) |
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| + | : Yes. If the kernel of <math>T</math> is non-trivial, then we have some non-zero <math>\vec{v}</math> with <math>T(\vec{v})=\vec{0}</math>. But also <math>T(\vec{0})=\vec{0}</math>, so <math>T</math> is not one-to-one and hence not invertible. [[User:Steven Glenn Jackson|Steven Glenn Jackson]] 15:27, 6 March 2012 (GMT) |
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| + | * If <math>\varphi:\mathbb{R}^n\to\mathbb{R}^n</math> is an isometry that fixes the origin: <math>\varphi(0) = 0</math>, then does <math>\varphi</math> preserve dot products? [[User:Patrickmclaren|Patrickmclaren]] 02:12, 4 April 2012 (BST) |
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| + | : Yes. Since <math>\phi</math> is an isometry and fixes the origin, it is linear (this is the only part that's hard to prove). Then since it preserves distance and fixes the origin, it preserves length of vectors. So it is an orthogonal transformation according to the book's definition. The book gives a proof that such transformations preserve dot products. [[User:Steven Glenn Jackson|Steven Glenn Jackson]] 13:52, 4 April 2012 (BST) |
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=Links= |
=Links= |
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* [http://www.wolframalpha.com/ WolframAlpha] |
* [http://www.wolframalpha.com/ WolframAlpha] |
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* [http://reference.wolfram.com/mathematica/guide/MatricesAndLinearAlgebra.html Matrices and Linear Algebra in Mathematica] |
* [http://reference.wolfram.com/mathematica/guide/MatricesAndLinearAlgebra.html Matrices and Linear Algebra in Mathematica] |
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| + | * [http://sagemath.org/ Sage], a free and open-source alternative to Mathematica and other mathematics software. (It can either be [http://sagemath.org/download-linux.html downloaded] or used online through a [http://www.sagenb.org/ web interface].) |
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| + | * A brief guide to [http://sagemath.org/doc/tutorial/tour_linalg.html linear algebra in Sage]. (Note that by default Sage uses row vectors and left kernels instead of column vectors and right kernels, so if you're not careful you may get into trouble with these functions. But its reduced row echelon forms are the same as ours.) |
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Latest revision as of 10:26, 4 May 2012
Math 260 --- Linear Algebra I
Welcome to the wiki! Editing this page is exactly like editing Wikipedia. You may wish to see their help pages on editing and on typesetting mathematics.
(It isn't as hard as the documentation might make it seem. To see how to typeset the sentence "Consider a linear transformation \(T:R^2\rightarrow R^2\)," click the "edit" link at the top of this page and read the source code that generated it.)
Steven Glenn Jackson 02:42, 1 March 2012 (UTC)
Important Dates[edit]
- 02/28/2012 - Exam 1
- 04/17/2012 - Exam 2
Questions[edit]
- If the kernel of a linear transformation is nontrivial, does that imply that the transformation is not injective and therefore the transformation matrix is not invertible? Patrickmclaren 02:11, 6 March 2012 (GMT)
- Yes. If the kernel of \(T\) is non-trivial, then we have some non-zero \(\vec{v}\) with \(T(\vec{v})=\vec{0}\). But also \(T(\vec{0})=\vec{0}\), so \(T\) is not one-to-one and hence not invertible. Steven Glenn Jackson 15:27, 6 March 2012 (GMT)
- If \(\varphi:\mathbb{R}^n\to\mathbb{R}^n\) is an isometry that fixes the origin\[\varphi(0) = 0\], then does \(\varphi\) preserve dot products? Patrickmclaren 02:12, 4 April 2012 (BST)
- Yes. Since \(\phi\) is an isometry and fixes the origin, it is linear (this is the only part that's hard to prove). Then since it preserves distance and fixes the origin, it preserves length of vectors. So it is an orthogonal transformation according to the book's definition. The book gives a proof that such transformations preserve dot products. Steven Glenn Jackson 13:52, 4 April 2012 (BST)
Links[edit]
- WolframAlpha
- Matrices and Linear Algebra in Mathematica
- Sage, a free and open-source alternative to Mathematica and other mathematics software. (It can either be downloaded or used online through a web interface.)
- A brief guide to linear algebra in Sage. (Note that by default Sage uses row vectors and left kernels instead of column vectors and right kernels, so if you're not careful you may get into trouble with these functions. But its reduced row echelon forms are the same as ours.)
