In [1]:

    

def vector_add(v, w):
    '''adds two vectors element by element'''
    return [v_i + w_i
           for v_i, w_i in zip(v,w)]


    

In [2]:

    

a = [1,2,3]
b = [5,6,7]


    

In [3]:

    

vector_add(a,b)


    

    
Out[3]:





[6, 8, 10]

In [4]:

    

def vector_subtract(v, w):
    '''subtracts element by element'''
    return [v_i - w_i
           for v_i, w_i in zip(v,w)]


    

In [5]:

    

vector_subtract(a,b)


    

    
Out[5]:





[-4, -4, -4]

In [6]:

    

# we can sum an arbitrary number of vectors with reduce
def vector_sum(vectors):
    return reduce(vector_add, vectors) # combine vectors with rule vector_add


    

In [7]:

    

vector_sum([a,b,a,b])


    

    
Out[7]:





[12, 16, 20]

In [8]:

    

def scalar_multiply(c, v):
    '''Takes scalar c and multiplies it by vector v'''
    return [c * v_i for v_i in v]


    

In [9]:

    

scalar_multiply(4,a)


    

    
Out[9]:





[4, 8, 12]

In [10]:

    

from __future__ import division
def vector_mean(vectors):
    '''elementwise mean of a collection of vectors'''
    n = len(vectors)
    return scalar_multiply(1/n, vector_sum(vectors))


    

In [11]:

    

vector_mean([a,b])


    

    
Out[11]:





[3.0, 4.0, 5.0]

In [12]:

    

def dot(v, w):
    '''We want the sum of the element wise products'''
    return sum(v_i * w_i
               for v_i,w_i in zip(v,w))


    

In [13]:

    

dot(a,b)


    

    
Out[13]:





38

In [14]:

    

def sum_of_squares(v):
    '''Simply the self dot product'''
    return dot(v,v)


    

In [15]:

    

sum_of_squares(a)


    

    
Out[15]:





14

In [16]:

    

import math

def magnitude(v):
    return math.sqrt(sum_of_squares(v))


    

In [17]:

    

magnitude(a)


    

    
Out[17]:





3.7416573867739413

In [18]:

    

# This is the sum of squares of the difference vector
def squared_distance(v, w):
    return sum_of_squares(vector_subtract(v, w))

# Which allows us to calc the distance
def distance(v,w):
    return math.sqrt(squared_distance(v, w))

# More directly, the magnitude of the difference vector
def distance(v,w):
    return magnitude(vector_subtract(v, w))


    

In [19]:

    

distance(a,b)


    

    
Out[19]:





6.928203230275509

In [20]:

    

# Shape
def shape(A):
    num_rows = len(A)
    num_cols = len(A[0]) if A else 0
    return num_rows, num_cols


    

In [21]:

    

A = [[1,2,3],
    [4,5,6]]

B = [[2,4],
    [6,8],
    [10,12]]


    

In [22]:

    

shape(A)


    

    
Out[22]:





(2, 3)

In [23]:

    

shape(B)


    

    
Out[23]:





(3, 2)

In [24]:

    

def get_row(A, i):
    return A[i]

def get_column(A, j):
    return [A_i[j]
           for A_i in A]


    

In [30]:

    

def make_matrix(num_rows, num_cols, make_fn):
    return [[make_fn(i, j)
            for j in xrange(num_cols)]
            for i in xrange(num_rows)]

def is_diagonal(i, j):
    '''1s on diag 0 otherwise'''
    return 1 if i == j else 0


    

In [31]:

    

make_matrix(4,4, is_diagonal)


    

    
Out[31]:





[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]

In [ ]: