NumPy [8] Álgebra lineal¶

In [ ]:

import numpy as np

m = np.array([[1,2,3], [-2,0,1]])
print(m)

[[ 1  2  3]
 [-2  0  1]]

In [ ]:

b = np.array([[1], [2], [3]])
print(b)

[[1]
 [2]
 [3]]

Transposición matricial¶

In [ ]:

a = np.transpose(b)           # cambiamos filas por columnas
print(a)

[[1 2 3]]

In [ ]:

np.transpose(m)

Out[ ]:

array([[ 1, -2],
       [ 2,  0],
       [ 3,  1]])

In [ ]:

print(np.transpose(a))

[[1]
 [2]
 [3]]

Es necesario usar dobles corchetes.

In [ ]:

print(np.transpose([[1, 2, 3]]))

[[1]
 [2]
 [3]]

$$Ax=b$$

Nuestro objetivo es calcular $x$.

$$A= \begin{pmatrix} 1 & -3 & 2\\ 5 & 6 & -1\\ 4 & -1 & 3 \end{pmatrix}$$

.

$$b= \begin{pmatrix} -3 \\ 13 \\ 8 \end{pmatrix}$$

Solución: $$x=A^{-1}b= \begin{pmatrix} -2 \\ 5 \\ 7 \end{pmatrix}$$

In [ ]:

A = np.array([[1,-3,2], [5,6,-1], [4,-1,3]])
print(A)

Out[ ]:

array([[ 1, -3,  2],
       [ 5,  6, -1],
       [ 4, -1,  3]])

In [ ]:

b = np.array([[-3], [13], [8]])
print(b)

[[-3]
 [13]
 [ 8]]

In [ ]:

x = np.linalg.solve(A, b)

In [ ]:

print(x)

[[-2.]
 [ 5.]
 [ 7.]]

La función allclose nos permite comprobar que los valores, aunque no sean exactamente iguales, si son muy cercanos.

In [ ]:

np.allclose(np.dot(A,x),b)

Out[ ]:

True

In [ ]:

np.dot(A, x)

Out[ ]:

array([[-3.],
       [13.],
       [ 8.]])

In [ ]:

Out[ ]:

array([[-3],
       [13],
       [ 8]])

In [ ]:

np.dot(A,x) - b

Out[ ]:

array([[-3.55271368e-15],
       [ 0.00000000e+00],
       [ 3.55271368e-15]])

En la práctica equivale a transponer el vector.

In [ ]:

c = np.array([1,2,3])
print(c)

[1 2 3]

In [ ]:

c.shape = (3, 1)
c

Out[ ]:

array([[1],
       [2],
       [3]])

In [ ]:

c.shape = (1, 3)
c

Out[ ]:

array([[1, 2, 3]])