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| Deletions are marked like this. | Additions are marked like this. |
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| by Marshall Hampton (tested by William Stein, who thinks this is really nice!) {{{ |
by Marshall Hampton {{{#!sagecell |
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| v = [a_list[i].copy() for i in indices] | v = [a_list[i][:] for i in indices] |
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| v = [a_list[i].copy() for i in indices] | v = [a_list[i][:] for i in indices] |
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| == Equality of det(A) and det(A.tranpose()) == srg = srange(-4,4,1/10,include_endpoint=True) @interact def dualv(a1=slider(srg,default=1),a2=slider(srg,default=2), a3=slider(srg,default=-1),a4=slider(srg,default=3)): A1 = arrow2d([0,0],[a1,a2],rgbcolor='black') A2 = arrow2d([0,0],[a3,a4],rgbcolor='black') A3 = arrow2d([0,0],[a1,a3],rgbcolor='black') A4 = arrow2d([0,0],[a2,a4],rgbcolor='black') p1 = polygon([[0,0],[a1,a2],[a1+a3,a2+a4],[a3,a4],[0,0]], alpha=.5) p2 = polygon([[0,0],[a1,a3],[a1+a2,a3+a4],[a2,a4],[0,0]],rgbcolor='red', alpha=.5) A = matrix([[a1,a2],[a3,a4]]) html('<h3>The determinant of a matrix is equal to the determinant of the transpose</h3>') html("$det(%s) = det(%s)$"%(latex(A),latex(A.transpose()))) show(A1+A2+A3+A4+p1+p2) |
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| {{{ @interact def linear_transformation(theta=slider(0, 2*pi, .1), r=slider(0.1, 2, .1, default=1)): A=matrix([[1,-1],[-1,1/2]]) |
{{{#!sagecell @interact def linear_transformation(A=matrix([[1,-1],[-1,1/2]]),theta=slider(0, 2*pi, .1), r=slider(0.1, 2, .1, default=1)): |
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| circles = sum([circle((0,0), radius=i, rgbcolor=(0,0,0)) for i in [1..2]]) print jsmath("v = %s,\; %s v=%s"%(v.n(4),latex(A),w.n(4))) show(v.plot(rgbcolor=(1,0,0))+w.plot(rgbcolor=(0,0,1))+circles,aspect_ratio=1) |
circles = sum([circle((0,0), radius=i, color='black') for i in [1..2]]) html("$%s %s=%s$"%tuple(map(latex, [A, v.column().n(4), w.column().n(4)]))) show(v.plot(color='red')+w.plot(color='blue')+circles,aspect_ratio=1) |
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| {{{ | {{{#!sagecell |
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| print jsmath('A = ' + latex(matrix(RealField(10),A))) | html('$A = ' + latex(matrix(RealField(10),A))+'$') |
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| {{{ | {{{#!sagecell |
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| def svd_vis(a11=slider(-1,1,.05,1),a12=slider(-1,1,.05,1),a21=slider(-1,1,.05,0),a22=slider(-1,1,.05,1),ofs= selector(['Off','On'],label='offset image from domain')): | def svd_vis(a11=slider(-1,1,.05,1),a12=slider(-1,1,.05,1),a21=slider(-1,1,.05,0),a22=slider(-1,1,.05,1),ofs= ('offset image from domain',False)): |
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| if ofs == 'On': | if ofs: |
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| print jsmath("$A = %s = %s %s %s$"%(latex(my_mat), latex(matrix(rf_low,u.tolist())), latex(matrix(rf_low,2,2,[s[0],0,0,s[1]])), latex(matrix(rf_low,vh.tolist())))) image_ell = parametric_plot(rotell(s,u,t, offset),0,2*pi) |
html("$A = %s = %s %s %s$"%(latex(my_mat), latex(matrix(rf_low,u.tolist())), latex(matrix(rf_low,2,2,[s[0],0,0,s[1]])), latex(matrix(rf_low,vh.tolist())))) image_ell = parametric_plot(rotell(s,u,t, offset),(0,2*pi)) |
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| show(graph_stuff,frame = False,axes=False,figsize=[fsize,fsize]) }}} |
show(graph_stuff,frame = False,axes=False,figsize=[fsize,fsize])}}} |
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| {{{ | {{{#!sagecell |
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| var('x') | |
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| show(plot(f, pbegin, pend, plot_points = 512), figsize = [4,3]) f_vals = [f(ind) for ind in srange(pbegin, pend,(pend-pbegin)/512.0)] |
show(plot(f, (x,pbegin, pend), plot_points = 512), figsize = [4,3]) f_vals = [f(x=ind) for ind in srange(pbegin, pend,(pend-pbegin)/512.0)] |
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| show(list_plot([abs(x) for x in my_fft], plotjoined=True), figsize = [4,3]) | show(list_plot([abs(i) for i in my_fft], plotjoined=True), figsize = [4,3]) |
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| {{{ | {{{#!sagecell |
Sage Interactions - Linear Algebra
goto interact main page
Contents
Numerical instability of the classical Gram-Schmidt algorithm
by Marshall Hampton
Equality of det(A) and det(A.tranpose())
srg = srange(-4,4,1/10,include_endpoint=True) @interact def dualv(a1=slider(srg,default=1),a2=slider(srg,default=2), a3=slider(srg,default=-1),a4=slider(srg,default=3)):
- A1 = arrow2d([0,0],[a1,a2],rgbcolor='black') A2 = arrow2d([0,0],[a3,a4],rgbcolor='black') A3 = arrow2d([0,0],[a1,a3],rgbcolor='black') A4 = arrow2d([0,0],[a2,a4],rgbcolor='black')
p1 = polygon(0,0],[a1,a2],[a1+a3,a2+a4],[a3,a4],[0,0, alpha=.5) p2 = polygon(0,0],[a1,a3],[a1+a2,a3+a4],[a2,a4],[0,0,rgbcolor='red', alpha=.5) A = matrix(a1,a2],[a3,a4) html('<h3>The determinant of a matrix is equal to the determinant of the transpose</h3>') html("det(%s) = det(%s)"%(latex(A),latex(A.transpose()))) show(A1+A2+A3+A4+p1+p2)
Linear transformations
by Jason Grout
A square matrix defines a linear transformation which rotates and/or scales vectors. In the interact command below, the red vector represents the original vector (v) and the blue vector represents the image w under the linear transformation. You can change the angle and length of v by changing theta and r.
Gerschgorin Circle Theorem
by Marshall Hampton. This animated version requires convert (imagemagick) to be installed, but it can easily be modified to a static version. The animation illustrates the idea behind the stronger version of Gerschgorin's theorem, which says that if the disks around the eigenvalues are disjoint then there is one eigenvalue per disk. The proof is by continuity of the eigenvalues under a homotopy to a diagonal matrix.
Singular value decomposition
by Marshall Hampton
Discrete Fourier Transform
by Marshall Hampton
The Gauss-Jordan method for inverting a matrix
by Hristo Inouzhe
...(goes all the way to invert the matrix)