Reverse-Engineering Linear Algebra Billy Wonderly Department of Mathematics and Computer Science University of Puget Sound Mathematics Seminar Septemb
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1 Department of Mathematics and Computer Science University of Puget Sound Mathematics Seminar September 13, 2010
2 Purpose for Research
3 Purpose for Research Write code in Sage that is useful to undergraduates interested in Linear Algebra.
4 Purpose for Research Write code in Sage that is useful to undergraduates interested in Linear Algebra. Generate problems with nice solutions.
5 Purpose for Research Write code in Sage that is useful to undergraduates interested in Linear Algebra. Generate problems with nice solutions RREF
6 What is Sage?
7 What is Sage? Sage is open-source mathematics software.
8 What is Sage? Sage is open-source mathematics software. Created by William Stein for the purpose of creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab.
9 What is Sage? Sage is open-source mathematics software. Created by William Stein for the purpose of creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab. Primarily developed for, and by, graduate or post-graduate mathematicians.
10 Approach
11 Approach Reverse-engineer a problem from a solution.
12 Approach Reverse-engineer a problem from a solution
13 Approach Reverse-engineer a problem from a solution Reverse GGGGGGGGGGGGA Engineer
14 Approach Reverse-engineer a problem from a solution Reverse GGGGGGGGGGGGA Engineer RREF
15 Summary of Routines The following are now possible inputs for the Sage random matrix() function.
16 Summary of Routines The following are now possible inputs for the Sage random matrix() function. echelon form: produces a matrix in reduced row-echelon form.
17 Summary of Routines The following are now possible inputs for the Sage random matrix() function. echelon form: produces a matrix in reduced row-echelon form. echelonizable: produces a matrix with a nice echelon form.
18 Summary of Routines The following are now possible inputs for the Sage random matrix() function. echelon form: produces a matrix in reduced row-echelon form. echelonizable: produces a matrix with a nice echelon form. unimodular: generates a matrix with determinant 1.
19 Summary of Routines The following are now possible inputs for the Sage random matrix() function. echelon form: produces a matrix in reduced row-echelon form. echelonizable: produces a matrix with a nice echelon form. unimodular: generates a matrix with determinant 1. subspaces: generates a matrix with nice natural basis vectors for its four fundamental subspaces.
20 Summary of Routines The following are now possible inputs for the Sage random matrix() function. echelon form: produces a matrix in reduced row-echelon form. echelonizable: produces a matrix with a nice echelon form. unimodular: generates a matrix with determinant 1. subspaces: generates a matrix with nice natural basis vectors for its four fundamental subspaces. diagonalizable: creates a diagonalizable matrix whose eigenvectors, if computed by hand, have only integer entries.
21 Matrices with Nice Echelon Form
22 Matrices with Nice Echelon Form Strategy: take a matrix in reduced row-echelon form and perform a reverse Gauss-Jordan elimination.
23 Matrices with Nice Echelon Form Strategy: take a matrix in reduced row-echelon form and perform a reverse Gauss-Jordan elimination
24 Matrices with Nice Echelon Form Strategy: take a matrix in reduced row-echelon form and perform a reverse Gauss-Jordan elimination row GGGGGGGGGA operations
25 Matrices with Nice Echelon Form Strategy: take a matrix in reduced row-echelon form and perform a reverse Gauss-Jordan elimination row GGGGGGGGGA operations
26 Matrices with Nice Echelon Form Strategy: take a matrix in reduced row-echelon form and perform a reverse Gauss-Jordan elimination row GGGGGGGGGA operations RREF
27 Diagonalizable Generate a diagonalizable matrix by designing a desirable matrix of eigenvectors and reversing the similarity transformation D = S 1 AS to give SDS 1 = A
28 Diagonalizable Generate a diagonalizable matrix by designing a desirable matrix of eigenvectors and reversing the similarity transformation D = S 1 AS to give SDS 1 = A
29 Diagonalizable Generate a diagonalizable matrix by designing a desirable matrix of eigenvectors and reversing the similarity transformation D = S 1 AS to give SDS 1 = A
30 Diagonalizable Generate a diagonalizable matrix by designing a desirable matrix of eigenvectors and reversing the similarity transformation D = S 1 AS to give SDS 1 = A
31 Diagonalizable Generate a diagonalizable matrix by designing a desirable matrix of eigenvectors and reversing the similarity transformation D = S 1 AS to give SDS 1 = A =
32 Sage Development Modify the Sage source code
33 Sage Development Modify the Sage source code Submit a patch by posting it on trac.sagemath.org as a ticket
34 Sage Development Modify the Sage source code Submit a patch by posting it on trac.sagemath.org as a ticket Wait for another developer to review the ticket and eventually mark as positive review
35 Sage Development Modify the Sage source code Submit a patch by posting it on trac.sagemath.org as a ticket Wait for another developer to review the ticket and eventually mark as positive review Prof. David Joyner, United States Naval Academy
36 Sage Development Modify the Sage source code Submit a patch by posting it on trac.sagemath.org as a ticket Wait for another developer to review the ticket and eventually mark as positive review Prof. David Joyner, United States Naval Academy Wait for the patch to be merged in the next release.
37 Conclusions Linear Algebra problems can be produced by generating solutions and working backwards.
38 Conclusions Linear Algebra problems can be produced by generating solutions and working backwards. Take Computer Science 161 early.
39 Conclusions Linear Algebra problems can be produced by generating solutions and working backwards. Take Computer Science 161 early. Open source development is fun
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