2015-11-17-120709.sagews

from sage.symbolic.integration.integral import indefinite_integral
#A4.4a+b
M3 = MatrixSpace(RR, 3) # Rational numbers
print("Identity matrix:")
show(M3.identity_matrix())
A = M3.matrix([[1, 0, 1],
               [-1, 0, 1], 
               [ 1, 2, 1]])

print("Matrix A:")
show(A)

b=vector([2,0,2/3])
print("Vector b:")
show(b)

B=A.augment(b)
show(B)
C=B.rref()
show(C)

#A4.4c

f(x)=exp(x)
plot(f, -2,2)
numerical_approx(f.integral(x, -2, 2))

#A4.4d
M5= MatrixSpace(RR, 5)
A = M5.matrix([[1, 0, 1, 0,2], [-1, 0, 1, 0,0], [1, 2, 1, 0,2/3],
[-1,0, 1, 0,0],[1,0,1,24,2/5]])
show(A.rref())

Identity matrix:

$\displaystyle \left(\begin{array}{rrr} 1.00000000000000 & 0.000000000000000 & 0.000000000000000 \\ 0.000000000000000 & 1.00000000000000 & 0.000000000000000 \\ 0.000000000000000 & 0.000000000000000 & 1.00000000000000 \end{array}\right)$

Matrix A:

$\displaystyle \left(\begin{array}{rrr} 1.00000000000000 & 0.000000000000000 & 1.00000000000000 \\ -1.00000000000000 & 0.000000000000000 & 1.00000000000000 \\ 1.00000000000000 & 2.00000000000000 & 1.00000000000000 \end{array}\right)$

Vector b:

$\displaystyle \left(2,\,0,\,\frac{2}{3}\right)$

$\displaystyle \left(\begin{array}{rrrr} 1.00000000000000 & 0.000000000000000 & 1.00000000000000 & 2.00000000000000 \\ -1.00000000000000 & 0.000000000000000 & 1.00000000000000 & 0.000000000000000 \\ 1.00000000000000 & 2.00000000000000 & 1.00000000000000 & 0.666666666666667 \end{array}\right)$

$\displaystyle \left(\begin{array}{rrrr} 1.00000000000000 & 0.000000000000000 & 0.000000000000000 & 1.00000000000000 \\ 0.000000000000000 & 1.00000000000000 & 0.000000000000000 & -0.666666666666667 \\ 0.000000000000000 & 0.000000000000000 & 1.00000000000000 & 1.00000000000000 \end{array}\right)$

7.25372081569404 Identity matrix:

$\displaystyle \left(\begin{array}{rrrrr} 1.00000000000000 & 0.000000000000000 & 0.000000000000000 & 0.000000000000000 & 0.000000000000000 \\ 0.000000000000000 & 1.00000000000000 & 0.000000000000000 & 0.000000000000000 & 0.000000000000000 \\ 0.000000000000000 & 0.000000000000000 & 1.00000000000000 & 0.000000000000000 & 0.000000000000000 \\ 0.000000000000000 & 0.000000000000000 & 0.000000000000000 & 1.00000000000000 & 0.000000000000000 \\ 0.000000000000000 & 0.000000000000000 & 0.000000000000000 & 0.000000000000000 & 1.00000000000000 \end{array}\right)$

$\displaystyle \left(\begin{array}{rrrrr} 1.00000000000000 & 0.000000000000000 & 0.000000000000000 & 0.000000000000000 & 1.00000000000000 \\ 0.000000000000000 & 1.00000000000000 & 0.000000000000000 & 0.000000000000000 & -0.666666666666667 \\ 0.000000000000000 & 0.000000000000000 & 1.00000000000000 & 0.000000000000000 & 1.00000000000000 \\ 0.000000000000000 & 0.000000000000000 & 0.000000000000000 & 1.00000000000000 & -0.0666666666666667 \\ 0.000000000000000 & 0.000000000000000 & 0.000000000000000 & 0.000000000000000 & 0.000000000000000 \end{array}\right)$