Is there anything more useless or less useful than Algebra?
Billy Connolly
Linear algebra is the branch of mathematics that deals with vector spaces. Although I can’t hope to teach you linear algebra in a brief chapter, it underpins a large number of data science concepts and techniques, which means I owe it to you to at least try. What we learn in this chapter we’ll use heavily throughout the rest of the book.
Abstractly, vectors are objects that can be added together (to form new vectors) and that can be multiplied by scalars (i.e., numbers), also to form new vectors.
Concretely (for us), vectors are points in some finitedimensional space. Although you might not think of your data as vectors, they are a good way to represent numeric data.
For example, if you have the heights, weights, and ages of a large number of people, you can treat your data as threedimensional vectors (height, weight, age)
. If you’re teaching a class with four exams, you can treat student grades as fourdimensional vectors (exam1, exam2, exam3, exam4)
.
The simplest fromscratch approach is to represent vectors as lists of numbers. A list of three numbers corresponds to a vector in threedimensional space, and vice versa:
height_weight_age
=
[
70
,
# inches,
170
,
# pounds,
40
]
# years
grades
=
[
95
,
# exam1
80
,
# exam2
75
,
# exam3
62
]
# exam4
One problem with this approach is that we will want to perform arithmetic on vectors. Because Python lists aren’t vectors (and hence provide no facilities for vector arithmetic), we’ll need to build these arithmetic tools ourselves. So let’s start with that.
To begin with, we’ll frequently need to add two vectors. Vectors add componentwise. This means that if two vectors v
and w
are the same length, their sum is just the vector whose first element is v[0] + w[0]
, whose second element is v[1] + w[1]
, and so on. (If they’re not the same length, then we’re not allowed to add them.)
For example, adding the vectors [1, 2]
and [2, 1]
results in [1 + 2, 2 + 1]
or [3, 3]
, as shown in Figure 41.
We can easily implement this by zip
ing the vectors together and using a list comprehension to add the corresponding elements:
def
vector_add
(
v
,
w
):
"""adds corresponding elements"""
return
[
v_i
+
w_i
for
v_i
,
w_i
in
zip
(
v
,
w
)]
Similarly, to subtract two vectors we just subtract corresponding elements:
def
vector_subtract
(
v
,
w
):
"""subtracts corresponding elements"""
return
[
v_i

w_i
for
v_i
,
w_i
in
zip
(
v
,
w
)]
We’ll also sometimes want to componentwise sum a list of vectors. That is, create a new vector whose first element is the sum of all the first elements, whose second element is the sum of all the second elements, and so on. The easiest way to do this is by adding one vector at a time:
def
vector_sum
(
vectors
):
"""sums all corresponding elements"""
result
=
vectors
[
0
]
# start with the first vector
for
vector
in
vectors
[
1
:]:
# then loop over the others
result
=
vector_add
(
result
,
vector
)
# and add them to the result
return
result
If you think about it, we are just reduce
ing the list of vectors using vector_add
, which means we can rewrite this more briefly using higherorder functions:
def
vector_sum
(
vectors
):
return
reduce
(
vector_add
,
vectors
)
or even:
vector_sum
=
partial
(
reduce
,
vector_add
)
although this last one is probably more clever than helpful.
We’ll also need to be able to multiply a vector by a scalar, which we do simply by multiplying each element of the vector by that number:
def
scalar_multiply
(
c
,
v
):
"""c is a number, v is a vector"""
return
[
c
*
v_i
for
v_i
in
v
]
This allows us to compute the componentwise means of a list of (samesized) vectors:
def
vector_mean
(
vectors
):
"""compute the vector whose ith element is the mean of the
ith elements of the input vectors"""
n
=
len
(
vectors
)
return
scalar_multiply
(
1
/
n
,
vector_sum
(
vectors
))
A less obvious tool is the dot product. The dot product of two vectors is the sum of their componentwise products:
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
))
The dot product measures how far the vector v extends in the w direction. For example, if w = [1, 0]
then dot(v, w)
is just the first component of v
. Another way of saying this is that it’s the length of the vector you’d get if you projected v onto w (Figure 42).
Using this, it’s easy to compute a vector’s sum of squares:
def
sum_of_squares
(
v
):
"""v_1 * v_1 + ... + v_n * v_n"""
return
dot
(
v
,
v
)
Which we can use to compute its magnitude (or length):
import
math
def
magnitude
(
v
):
return
math
.
sqrt
(
sum_of_squares
(
v
))
# math.sqrt is square root function
We now have all the pieces we need to compute the distance between two vectors, defined as:
def
squared_distance
(
v
,
w
):
"""(v_1  w_1) ** 2 + ... + (v_n  w_n) ** 2"""
return
sum_of_squares
(
vector_subtract
(
v
,
w
))
def
distance
(
v
,
w
):
return
math
.
sqrt
(
squared_distance
(
v
,
w
))
Which is possibly clearer if we write it as (the equivalent):
def
distance
(
v
,
w
):
return
magnitude
(
vector_subtract
(
v
,
w
))
That should be plenty to get us started. We’ll be using these functions heavily throughout the book.
A matrix is a twodimensional collection of numbers. We will represent matrices as list
s of list
s, with each inner list having the same size and representing a row of the matrix. If A
is a matrix, then A[i][j]
is the element in the ith row and the jth column. Per mathematical convention, we will typically use capital letters to represent matrices. For example:
A
=
[[
1
,
2
,
3
],
# A has 2 rows and 3 columns
[
4
,
5
,
6
]]
B
=
[[
1
,
2
],
# B has 3 rows and 2 columns
[
3
,
4
],
[
5
,
6
]]
In mathematics, you would usually name the first row of the matrix “row 1” and the first column “column 1.” Because we’re representing matrices with Python list
s, which are zeroindexed, we’ll call the first row of a matrix “row 0” and the first column “column 0.”
Given this listoflists representation, the matrix A
has len(A)
rows and len(A[0])
columns, which we consider its shape
:
def
shape
(
A
):
num_rows
=
len
(
A
)
num_cols
=
len
(
A
[
0
])
if
A
else
0
# number of elements in first row
return
num_rows
,
num_cols
If a matrix has n rows and k columns, we will refer to it as a matrix. We can (and sometimes will) think of each row of a matrix as a vector of length k, and each column as a vector of length n:
def
get_row
(
A
,
i
):
return
A
[
i
]
# A[i] is already the ith row
def
get_column
(
A
,
j
):
return
[
A_i
[
j
]
# jth element of row A_i
for
A_i
in
A
]
# for each row A_i
We’ll also want to be able to create a matrix given its shape and a function for generating its elements. We can do this using a nested list comprehension:
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
)
# given i, create a list
for
j
in
range
(
num_cols
)]
# [entry_fn(i, 0), ... ]
for
i
in
range
(
num_rows
)]
# create one list for each i
Given this function, you could make a 5 × 5 identity matrix (with 1s on the diagonal and 0s elsewhere) with:
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
)
# [[1, 0, 0, 0, 0],
# [0, 1, 0, 0, 0],
# [0, 0, 1, 0, 0],
# [0, 0, 0, 1, 0],
# [0, 0, 0, 0, 1]]
Matrices will be important to us for several reasons.
First, we can use a matrix to represent a data set consisting of multiple vectors, simply by considering each vector as a row of the matrix. For example, if you had the heights, weights, and ages of 1,000 people you could put them in a matrix:
data
=
[[
70
,
170
,
40
],
[
65
,
120
,
26
],
[
77
,
250
,
19
],
# ....
]
Second, as we’ll see later, we can use an matrix to represent a linear function that maps kdimensional vectors to ndimensional vectors. Several of our techniques and concepts will involve such functions.
Third, matrices can be used to represent binary relationships. In Chapter 1, we represented the edges of a network as a collection of pairs (i, j)
. An alternative representation would be to create a matrix A such that A[i][j]
is 1 if nodes i and j are connected and 0 otherwise.
Recall that before we had:
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
)]
We could also represent this as:
# 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
If there are very few connections, this is a much more inefficient representation, since you end up having to store a lot of zeroes. However, with the matrix representation it is much quicker to check whether two nodes are connected—you just have to do a matrix lookup instead of (potentially) inspecting every edge:
friendships
[
0
][
2
]
==
1
# True, 0 and 2 are friends
friendships
[
0
][
8
]
==
1
# False, 0 and 8 are not friends
Similarly, to find the connections a node has, you only need to inspect the column (or the row) corresponding to that node:
friends_of_five
=
[
i
# only need
for
i
,
is_friend
in
enumerate
(
friendships
[
5
])
# to look at
if
is_friend
]
# one row
Previously we added a list of connections to each node object to speed up this process, but for a large, evolving graph that would probably be too expensive and difficult to maintain.
Linear algebra is widely used by data scientists (frequently implicitly, and not infrequently by people who don’t understand it). It wouldn’t be a bad idea to read a textbook. You can find several freely available online:
If you are feeling adventurous, Linear Algebra Done Wrong is a more advanced introduction
All of the machinery we built here you get for free if you use NumPy. (You get a lot more too.)
No credit card required