-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathlinear_algebra.py
148 lines (114 loc) · 4.49 KB
/
linear_algebra.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
#!/usr/bin/python
# -*- coding: utf-8 -*-
# Copyright (C) 2018 David Arroyo Menéndez
# Author: David Arroyo Menéndez <[email protected]>
# Maintainer: David Arroyo Menéndez <[email protected]>
# This file is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 3, or (at your option)
# any later version.
# This file is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
# You should have received a copy of the GNU General Public License
# along with GNU Emacs; see the file COPYING. If not, write to
# the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
# Boston, MA 02110-1301 USA,
# -*- coding: iso-8859-15 -*-
from __future__ import division # want 3 / 2 == 1.5
import re, math, random # regexes, math functions, random numbers
import matplotlib.pyplot as plt # pyplot
from collections import defaultdict, Counter
from functools import partial
#
# functions for working with vectors
#
def vector_add(v, w):
"""adds two vectors componentwise"""
return [v_i + w_i for v_i, w_i in zip(v,w)]
def vector_subtract(v, w):
"""subtracts two vectors componentwise"""
return [v_i - w_i for v_i, w_i in zip(v,w)]
def vector_sum(vectors):
return reduce(vector_add, vectors)
def scalar_multiply(c, v):
return [c * v_i for v_i in v]
# this isn't right if you don't from __future__ import division
def vector_mean(vectors):
"""compute the vector whose i-th element is the mean of the
i-th elements of the input vectors"""
n = len(vectors)
return scalar_multiply(1/n, vector_sum(vectors))
def dot(v, w):
"""v_1 * w_1 + ... + v_n * w_n"""
return sum(v_i * w_i for v_i, w_i in zip(v, w))
def sum_of_squares(v):
"""v_1 * v_1 + ... + v_n * v_n"""
return dot(v, v)
def magnitude(v):
return math.sqrt(sum_of_squares(v))
def squared_distance(v, w):
return sum_of_squares(vector_subtract(v, w))
def distance(v, w):
return math.sqrt(squared_distance(v, w))
#
# functions for working with matrices
#
def shape(A):
num_rows = len(A)
num_cols = len(A[0]) if A else 0
return num_rows, num_cols
def get_row(A, i):
return A[i]
def get_column(A, j):
return [A_i[j] for A_i in A]
def make_matrix(num_rows, num_cols, entry_fn):
"""returns a num_rows x num_cols matrix
whose (i,j)-th entry is entry_fn(i, j)"""
return [[entry_fn(i, j) for j in range(num_cols)]
for i in range(num_rows)]
def is_diagonal(i, j):
"""1's on the 'diagonal', 0's everywhere else"""
return 1 if i == j else 0
identity_matrix = make_matrix(5, 5, is_diagonal)
# user 0 1 2 3 4 5 6 7 8 9
#
friendships = [[0, 1, 1, 0, 0, 0, 0, 0, 0, 0], # user 0
[1, 0, 1, 1, 0, 0, 0, 0, 0, 0], # user 1
[1, 1, 0, 1, 0, 0, 0, 0, 0, 0], # user 2
[0, 1, 1, 0, 1, 0, 0, 0, 0, 0], # user 3
[0, 0, 0, 1, 0, 1, 0, 0, 0, 0], # user 4
[0, 0, 0, 0, 1, 0, 1, 1, 0, 0], # user 5
[0, 0, 0, 0, 0, 1, 0, 0, 1, 0], # user 6
[0, 0, 0, 0, 0, 1, 0, 0, 1, 0], # user 7
[0, 0, 0, 0, 0, 0, 1, 1, 0, 1], # user 8
[0, 0, 0, 0, 0, 0, 0, 0, 1, 0]] # user 9
#####
# DELETE DOWN
#
def matrix_add(A, B):
if shape(A) != shape(B):
raise ArithmeticError("cannot add matrices with different shapes")
num_rows, num_cols = shape(A)
def entry_fn(i, j): return A[i][j] + B[i][j]
return make_matrix(num_rows, num_cols, entry_fn)
def make_graph_dot_product_as_vector_projection(plt):
v = [2, 1]
w = [math.sqrt(.25), math.sqrt(.75)]
c = dot(v, w)
vonw = scalar_multiply(c, w)
o = [0,0]
plt.arrow(0, 0, v[0], v[1],
width=0.002, head_width=.1, length_includes_head=True)
plt.annotate("v", v, xytext=[v[0] + 0.1, v[1]])
plt.arrow(0 ,0, w[0], w[1],
width=0.002, head_width=.1, length_includes_head=True)
plt.annotate("w", w, xytext=[w[0] - 0.1, w[1]])
plt.arrow(0, 0, vonw[0], vonw[1], length_includes_head=True)
plt.annotate(u"(v•w)w", vonw, xytext=[vonw[0] - 0.1, vonw[1] + 0.1])
plt.arrow(v[0], v[1], vonw[0] - v[0], vonw[1] - v[1],
linestyle='dotted', length_includes_head=True)
plt.scatter(*zip(v,w,o),marker='.')
plt.axis('equal')
plt.show()