Chapter 4. Linear Algebra

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.

Vectors

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 finite-dimensional space. Although you might not think of your data as vectors, they are often a useful 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 three-dimensional vectors [height, weight, age]. If you’re teaching a class with four exams, you can treat student grades as four-dimensional vectors [exam1, exam2, exam3, exam4].

The simplest from-scratch approach is to represent vectors as lists of numbers. A list of three numbers corresponds to a vector in three-dimensional space, and vice versa.

We’ll accomplish this with a type alias that says a Vector is just a list of floats:

from typing import List

Vector = List[float]

height_weight_age = [70,  # inches,
                     170, # pounds,
                     40 ] # years

grades = [95,   # exam1
          80,   # exam2
          75,   # exam3
          62 ]  # exam4

We’ll also 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 4-1.

Adding two vectors.
Figure 4-1. Adding two vectors

We can easily implement this by zip-ing the vectors together and using a list comprehension to add the corresponding elements:

def add(v: Vector, w: Vector) -> Vector:
    """Adds corresponding elements"""
    assert len(v) == len(w), "vectors must be the same length"

    return [v_i + w_i for v_i, w_i in zip(v, w)]

assert add([1, 2, 3], [4, 5, 6]) == [5, 7, 9]

Similarly, to subtract two vectors we just subtract the corresponding elements:

def subtract(v: Vector, w: Vector) -> Vector:
    """Subtracts corresponding elements"""
    assert len(v) == len(w), "vectors must be the same length"

    return [v_i - w_i for v_i, w_i in zip(v, w)]

assert subtract([5, 7, 9], [4, 5, 6]) == [1, 2, 3]

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:

def vector_sum(vectors: List[Vector]) -> Vector:
    """Sums all corresponding elements"""
    # Check that vectors is not empty
    assert vectors, "no vectors provided!"

    # Check the vectors are all the same size
    num_elements = len(vectors[0])
    assert all(len(v) == num_elements for v in vectors), "different sizes!"

    # the i-th element of the result is the sum of every vector[i]
    return [sum(vector[i] for vector in vectors)
            for i in range(num_elements)]

assert vector_sum([[1, 2], [3, 4], [5, 6], [7, 8]]) == [16, 20]

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: float, v: Vector) -> Vector:
    """Multiplies every element by c"""
    return [c * v_i for v_i in v]

assert scalar_multiply(2, [1, 2, 3]) == [2, 4, 6]

This allows us to compute the componentwise means of a list of (same-sized) vectors:

def vector_mean(vectors: List[Vector]) -> Vector:
    """Computes the element-wise average"""
    n = len(vectors)
    return scalar_multiply(1/n, vector_sum(vectors))

assert vector_mean([[1, 2], [3, 4], [5, 6]]) == [3, 4]

A less obvious tool is the dot product. The dot product of two vectors is the sum of their componentwise products:

def dot(v: Vector, w: Vector) -> float:
    """Computes v_1 * w_1 + ... + v_n * w_n"""
    assert len(v) == len(w), "vectors must be same length"

    return sum(v_i * w_i for v_i, w_i in zip(v, w))

assert dot([1, 2, 3], [4, 5, 6]) == 32  # 1 * 4 + 2 * 5 + 3 * 6

If w has magnitude 1, 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 4-2).

Dot product as a vector projection.
Figure 4-2. The dot product as vector projection

Using this, it’s easy to compute a vector’s sum of squares:

def sum_of_squares(v: Vector) -> float:
    """Returns v_1 * v_1 + ... + v_n * v_n"""
    return dot(v, v)

assert sum_of_squares([1, 2, 3]) == 14  # 1 * 1 + 2 * 2 + 3 * 3

which we can use to compute its magnitude (or length):

import math

def magnitude(v: Vector) -> float:
    """Returns the magnitude (or length) of v"""
    return math.sqrt(sum_of_squares(v))   # math.sqrt is square root function

assert magnitude([3, 4]) == 5

We now have all the pieces we need to compute the distance between two vectors, defined as:

(v 1 -w 1 ) 2 + ... + (v n -w n ) 2

In code:

def squared_distance(v: Vector, w: Vector) -> float:
    """Computes (v_1 - w_1) ** 2 + ... + (v_n - w_n) ** 2"""
    return sum_of_squares(subtract(v, w))

def distance(v: Vector, w: Vector) -> float:
    """Computes the distance between v and w"""
    return math.sqrt(squared_distance(v, w))

This is possibly clearer if we write it as (the equivalent):

def distance(v: Vector, w: Vector) -> float:
    return magnitude(subtract(v, w))

That should be plenty to get us started. We’ll be using these functions heavily throughout the book.

Note

Using lists as vectors is great for exposition but terrible for performance.

In production code, you would want to use the NumPy library, which includes a high-performance array class with all sorts of arithmetic operations included.

Matrices

A matrix is a two-dimensional collection of numbers. We will represent matrices as lists of lists, 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 frequently use capital letters to represent matrices. For example:

# Another type alias
Matrix = List[List[float]]

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]]
Note

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 lists, which are zero-indexed, we’ll call the first row of a matrix “row 0” and the first column “column 0.”

Given this list-of-lists representation, the matrix A has len(A) rows and len(A[0]) columns, which we consider its shape:

from typing import Tuple

def shape(A: Matrix) -> Tuple[int, int]:
    """Returns (# of rows of A, # of columns of 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

assert shape([[1, 2, 3], [4, 5, 6]]) == (2, 3)  # 2 rows, 3 columns

If a matrix has n rows and k columns, we will refer to it as an n × k matrix. We can (and sometimes will) think of each row of an n × k matrix as a vector of length k, and each column as a vector of length n:

def get_row(A: Matrix, i: int) -> Vector:
    """Returns the i-th row of A (as a Vector)"""
    return A[i]             # A[i] is already the ith row

def get_column(A: Matrix, j: int) -> Vector:
    """Returns the j-th column of A (as a Vector)"""
    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:

from typing import Callable

def make_matrix(num_rows: int,
                num_cols: int,
                entry_fn: Callable[[int, int], float]) -> Matrix:
    """
    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) like so:

def identity_matrix(n: int) -> Matrix:
    """Returns the n x n identity matrix"""
    return make_matrix(n, n, lambda i, j: 1 if i == j else 0)

assert identity_matrix(5) == [[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 dataset 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 1,000 × 3 matrix:

data = [[70, 170, 40],
        [65, 120, 26],
        [77, 250, 19],
        # ....
       ]

Second, as we’ll see later, we can use an n × k matrix to represent a linear function that maps k-dimensional vectors to n-dimensional 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
#
friend_matrix = [[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 zeros. 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:

assert friend_matrix[0][2] == 1, "0 and 2 are friends"
assert friend_matrix[0][8] == 0, "0 and 8 are not friends"

Similarly, to find a node’s connections, you only need to inspect the column (or the row) corresponding to that node:

# only need to look at one row
friends_of_five = [i
                   for i, is_friend in enumerate(friend_matrix[5])
                   if is_friend]

With a small graph you could just add 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.

We’ll revisit matrices throughout the book.

For Further Exploration

  • 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:

    • Linear Algebra, by Jim Hefferon (Saint Michael’s College)

    • Linear Algebra, by David Cherney, Tom Denton, Rohit Thomas, and Andrew Waldron (UC Davis)

    • If you are feeling adventurous, Linear Algebra Done Wrong, by Sergei Treil (Brown University), is a more advanced introduction.

  • All of the machinery we built in this chapter you get for free if you use NumPy. (You get a lot more too, including much better performance.)

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