In [1]:
import re, math, random # regexes, math functions, random numbers
import matplotlib.pyplot as plt # pyplot
from collections import defaultdict, Counter
from functools import partial, reduce
In [19]:
v = [1, 2]
w = [2, 1]
vectors = [v, w]
In [2]:
def vector_add(v, w):
"""adds two vectors componentwise"""
return [v_i + w_i for v_i, w_i in zip(v,w)]
In [3]:
vector_add(v, w)
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In [5]:
def vector_subtract(v, w):
"""subtracts two vectors componentwise"""
return [v_i - w_i for v_i, w_i in zip(v,w)]
vector_subtract(v, w)
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In [7]:
def vector_sum(vectors):
return reduce(vector_add, vectors)
In [20]:
vector_sum(vectors)
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In [2]:
def scalar_multiply(c, v):
# c is a number, v is a vector
return [c * v_i for v_i in v]
In [13]:
scalar_multiply(2.5, v)
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In [14]:
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))
In [21]:
vector_mean(vectors)
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In [4]:
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))
In [18]:
dot(v, w)
Out[18]:
The dot product measures how far the vector v extends in the w direction.
The dot product measures the length of the vector you’d get if you projected v onto w.
In [22]:
def sum_of_squares(v):
"""v_1 * v_1 + ... + v_n * v_n"""
return dot(v, v)
In [23]:
sum_of_squares(v)
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In [24]:
def magnitude(v):
return math.sqrt(sum_of_squares(v))
In [25]:
magnitude(v)
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In [26]:
def squared_distance(v, w):
return sum_of_squares(vector_subtract(v, w))
In [27]:
squared_distance(v, w)
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In [28]:
def distance(v, w):
return math.sqrt(squared_distance(v, w))
In [29]:
distance(v, w)
Out[29]:
Using lists as vectors
A matrix is a two-dimensional collection of numbers.
In [30]:
A = [[1, 2, 3],
[4, 5, 6]]
B = [[1, 2],
[3, 4],
[5, 6]]
In [31]:
def shape(A):
num_rows = len(A)
num_cols = len(A[0]) if A else 0
return num_rows, num_cols
In [32]:
shape(A)
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In [34]:
def get_row(A, i):
return A[i]
In [35]:
get_row(A, 1)
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In [37]:
def get_column(A, j):
return [A_i[j] for A_i in A]
In [38]:
get_column(A, 2)
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In [50]:
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),
entry_fn is a function for generating matrix elements."""
return [[entry_fn(i, j)
for j in range(num_cols)]
for i in range(num_rows)]
In [71]:
def entry_add(i, j):
"""a function for generating matrix elements. """
return i+j
make_matrix(5, 5, entry_add)
Out[71]:
In [53]:
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)
identity_matrix
Out[53]:
In [58]:
friendships = [(0, 1),
(0, 2),
(1, 2),
(1, 3),
(2, 3),
(3, 4),
(4, 5),
(5, 6),
(5, 7),
(6, 8),
(7, 8),
(8, 9)]
In [60]:
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
In [62]:
friendships[0][2] == 1 # True, 0 and 2 are friends
Out[62]:
In [74]:
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)
In [76]:
A = make_matrix(5, 5, is_diagonal)
B = make_matrix(5, 5, entry_add)
matrix_add(A, B)
Out[76]:
In [104]:
v = [2, 1]
w = [math.sqrt(.25), math.sqrt(.75)]
c = dot(v, w)
vonw = scalar_multiply(c, w)
o = [0,0]
plt.figure(figsize=(4, 5), dpi = 100)
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] + 0.01, 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] + 0.02])
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()