2+3
5 5 |
\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}=\prod_p \left(\frac{1}{1-p^{-s}}\right)
Matrix([[1,2,3],[3,2,1],[1,1,1]]); A
[1 2 3] [3 2 1] [1 1 1] Traceback (click to the left of this block for traceback) ... NameError: name 'A' is not defined [1 2 3]
[3 2 1]
[1 1 1]
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_2.py", line 5, in <module>
Matrix([[_sage_const_1 ,_sage_const_2 ,_sage_const_3 ],[_sage_const_3 ,_sage_const_2 ,_sage_const_1 ],[_sage_const_1 ,_sage_const_1 ,_sage_const_1 ]]); A
File "", line 1, in <module>
NameError: name 'A' is not defined
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A = Matrix([[1,2,3],[3,2,1],[1,1,1]])
A
[1 2 3] [3 2 1] [1 1 1] [1 2 3] [3 2 1] [1 1 1] |
Matrix([[1,2,3],[3,2,1],[1,1,1]]); A
[1 2 3] [3 2 1] [1 1 1] [1 2 3] [3 2 1] [1 1 1] [1 2 3] [3 2 1] [1 1 1] [1 2 3] [3 2 1] [1 1 1] |
w = vector([1,1,-4]); w; w*A; A*w
(1, 1, -4) (0, 0, 0) (-9, 1, -2) (1, 1, -4) (0, 0, 0) (-9, 1, -2) |
Y = vector([0,-4,-1])
X = A.solve_right(Y)
A \ Y; X; A*X
(-2, 1, 0) (-2, 1, 0) (0, -4, -1) (-2, 1, 0) (-2, 1, 0) (0, -4, -1) |
A.solve_right(w)
Traceback (click to the left of this block for traceback) ... ValueError: matrix equation has no solutions Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_8.py", line 4, in <module>
A.solve_right(w)
File "", line 1, in <module>
File "matrix2.pyx", line 280, in sage.matrix.matrix2.Matrix.solve_right (sage/matrix/matrix2.c:3808)
File "matrix2.pyx", line 400, in sage.matrix.matrix2.Matrix._solve_right_general (sage/matrix/matrix2.c:4609)
ValueError: matrix equation has no solutions
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A.echelon_form();A.determinant();A.rank()
[ 1 0 -1] [ 0 1 2] [ 0 0 0] 0 2 [ 1 0 -1] [ 0 1 2] [ 0 0 0] 0 2 |
echelon_form(A)
Traceback (click to the left of this block for traceback) ... NameError: name 'echelon_form' is not defined Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_10.py", line 4, in <module>
echelon_form(A)
File "", line 1, in <module>
NameError: name 'echelon_form' is not defined
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A.eigenspaces_left?
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A.eigenspaces_left()
[ (5, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [1 1 1]), (0, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [ 1 1 -4]), (-1, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [ 1 -1/5 -7/5]) ] [ (5, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [1 1 1]), (0, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [ 1 1 -4]), (-1, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [ 1 -1/5 -7/5]) ] |
y = var('y')
P=plot_slope_field(1-y,(x,0,3),(y,0,20))
y = function('y',x) # declare y to be a function of x
f = desolve(diff(y,x) + y - 1, y, ics=[2,2]) # solve the DE, with Initial ConditionS of 2,2
Q=plot(f,0,3)
html('$f=%s$'%(latex(f),)); show(P+Q) # I can use HTML in my outcomes
f={(e^{2} + e^{x})} e^{-x} f={(e^{2} + e^{x})} e^{-x} |
f(x)=x^2; f(3)
9 9 |
plot([x^i for i in [1..10]],(x,0,1),fill = dict((i,[i+1]) for i in [0..9]))
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import pylab
A_image = pylab.mean(pylab.imread(DATA + 'WNECMath.png'), 2)
@interact
def svd_image(i=(20,(1..100)),display_axes=True):
u,s,v = pylab.linalg.svd(A_image)
A = sum(s[j]*pylab.outer(u[0:,j],v[j,0:]) for j in range(i))
g = graphics_array([matrix_plot(A),matrix_plot(A_image)])
show(g,axes=display_axes, figsize=(8,3))
html('<h2>Compressed using %s eigenvalues</h2>'%i)
Click to the left again to hide and once more to show the dynamic interactive window |
var('x,y')
plot3d(sin(pi*(x^2+y^2))/2,(x,-1,1),(y,-5,5))
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