Course Content

Ultimate NumPy

Basic Linear Algebra with NumPy

Linear algebra is a fundamental branch of mathematics that plays a crucial role in various fields, including machine learning, deep learning, and data analysis.

Vectors and Matrices

In linear algebra, a vector is an ordered set of values. 1D NumPy arrays can efficiently represent vectors. A matrix is a two-dimensional array of numbers, which can be represented by a 2D array in NumPy.

We have already covered vector and matrix addition and subtraction, as well as scalar multiplication, in the "Basic Mathematical Operations" chapter. Here, we will focus on other operations.

Transposition

Transposition is an operation that flips a matrix over its diagonal. In other words, it converts the rows of the matrix into columns and the columns into rows.

You can transpose a matrix using the .T attribute of a NumPy array:


              12345
            
import numpy as np
matrix = np.array([[1, 2, 3], [4, 5, 6]])
# Transposing a matrix
transposed_matrix = matrix.T
print(transposed_matrix)

Dot Product

The dot product is perhaps the most commonly used linear algebra operation in machine and deep learning. The dot product of two vectors (which must have an equal number of elements) is the sum of their element-wise products. The result is a scalar:

Matrix Multiplication

Matrix multiplication is defined only if the number of columns in the first matrix is equal to the number of rows in the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

As you can see, every element of the resulting matrix is the dot product of two vectors. The row number of the element corresponds to the number of the row vector in the first matrix, and the column number corresponds to the number of the column vector in the second matrix.

The number of columns in the first matrix must be equal to the number of rows in the second matrix, as the dot product requires the two vectors to have the same number of elements.

Dot Product and Matrix Multiplication in NumPy

NumPy provides the dot() function for both the dot product and matrix multiplication. This function takes two arrays as its arguments.

However, you can also use the @ operator between two arrays to achieve the same results.


              12345678910111213
            
import numpy as np
vector_1 = np.array([1, 2, 3])
vector_2 = np.array([4, 5, 6])
# Dot product using the dot() function
print(np.dot(vector_1, vector_2))
# Dot product using the @ operator
print(vector_1 @ vector_2)
matrix_1 = np.array([[1, 2, 3], [4, 5, 6]])
matrix_2 = np.array([[7, 10], [8, 11], [9, 12]])
# Matrix multiplication using the dot() function
print(np.dot(matrix_1, matrix_2))
# Matrix multiplication using the @ operator
print(matrix_1 @ matrix_2)

If the right argument in matrix multiplication is a vector (1D array), NumPy treats it as a matrix where the last dimension is 1. For example, when multiplying a 6x4 matrix by a vector with 4 elements, the vector is considered a 4x1 matrix.

If the left argument in matrix multiplication is a vector, NumPy treats it as a matrix where the first dimension is 1. For instance, when multiplying a vector with 4 elements by a 4x6 matrix, the vector is treated as a 1x4 matrix.

The picture below shows the structure of the exam_scores and coefficients arrays used in the task:

Task

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You are working with the exam_scores array, which contains simulated exam scores of three students (each row represents a student) across three subjects (each column represents a subject).

Multiply the scores of each subject exam by the respective coefficient.
Add the resulting scores for each student to calculate their final score.
Compute the dot product between exam_scores and coefficients.

This will give you the final scores for all students based on the weighted contributions of their subject scores.

Switch to desktop for real-world practiceContinue from where you are using one of the options below

Everything was clear?

Thanks for your feedback!

Section 4. Chapter 4

Basic Linear Algebra with NumPy

Linear algebra is a fundamental branch of mathematics that plays a crucial role in various fields, including machine learning, deep learning, and data analysis.

Vectors and Matrices

We have already covered vector and matrix addition and subtraction, as well as scalar multiplication, in the "Basic Mathematical Operations" chapter. Here, we will focus on other operations.

Transposition

Transposition is an operation that flips a matrix over its diagonal. In other words, it converts the rows of the matrix into columns and the columns into rows.

You can transpose a matrix using the .T attribute of a NumPy array:


              12345
            
import numpy as np
matrix = np.array([[1, 2, 3], [4, 5, 6]])
# Transposing a matrix
transposed_matrix = matrix.T
print(transposed_matrix)

Dot Product

Matrix Multiplication

The number of columns in the first matrix must be equal to the number of rows in the second matrix, as the dot product requires the two vectors to have the same number of elements.

Dot Product and Matrix Multiplication in NumPy

NumPy provides the dot() function for both the dot product and matrix multiplication. This function takes two arrays as its arguments.

However, you can also use the @ operator between two arrays to achieve the same results.


              12345678910111213
            
import numpy as np
vector_1 = np.array([1, 2, 3])
vector_2 = np.array([4, 5, 6])
# Dot product using the dot() function
print(np.dot(vector_1, vector_2))
# Dot product using the @ operator
print(vector_1 @ vector_2)
matrix_1 = np.array([[1, 2, 3], [4, 5, 6]])
matrix_2 = np.array([[7, 10], [8, 11], [9, 12]])
# Matrix multiplication using the dot() function
print(np.dot(matrix_1, matrix_2))
# Matrix multiplication using the @ operator
print(matrix_1 @ matrix_2)

The picture below shows the structure of the exam_scores and coefficients arrays used in the task:

Task

Swipe to show code editor

You are working with the exam_scores array, which contains simulated exam scores of three students (each row represents a student) across three subjects (each column represents a subject).

Multiply the scores of each subject exam by the respective coefficient.
Add the resulting scores for each student to calculate their final score.
Compute the dot product between exam_scores and coefficients.

This will give you the final scores for all students based on the weighted contributions of their subject scores.

Switch to desktop for real-world practiceContinue from where you are using one of the options below

Everything was clear?

Thanks for your feedback!

Section 4. Chapter 4

Switch to desktop for real-world practiceContinue from where you are using one of the options below