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The NumPy library in Python is widely used for linear algebra due to its powerful array handling capabilities. It provides efficient functions for a range of linear algebra operations, such as matrix creation, matrix operations, determinants, eigenvalues, and linear equation solving.
Vertically and Horizontally Stack Arrays with vstack and hstack
The np.array() function is used to create matrices (2D arrays) and vectors (1D arrays).
import numpy as np
# Creating a vector
vector = np.array([1, 2, 3])
# Creating a 2x2 matrix
matrix = np.array([[1, 2], [3, 4]])
# Creating a 3x3 matrix with zeros
zero_matrix = np.zeros((3, 3))
# Creating a 3x3 identity matrix
identity_matrix = np.eye(3)
| [Python NumPy Array Creation Methods Explained | Zeros, Ones, Empty, Arange, and Linspace](https://youtu.be/PI4UegrAYxs?si=v4imyXtI7YyBwH2Z) |
Basic arithmetic operations are easy with NumPy arrays and can be performed element-wise or with specific matrix functions.
np.dot() or the @ operator. A = np.array([[1, 2], [3, 4]])
B = np.array([[2, 0], [1, 3]])
# Element-wise addition and subtraction
addition = A + B
subtraction = A - B
# Scalar multiplication
scalar_multiplication = 2 * A
# Matrix multiplication
matrix_multiplication = np.dot(A, B) # or A @ B
Transposing a matrix (switching rows and columns) is straightforward with the .T attribute.
A = np.array([[1, 2], [3, 4]])
transpose_A = A.T
The determinant is a scalar value that can be computed using np.linalg.det(), which is useful in solving linear systems and understanding matrix properties.
A = np.array([[1, 2], [3, 4]])
det_A = np.linalg.det(A) # Output: -2.0
The inverse of a matrix ( A ) is another matrix ( A^{-1} ) such that ( A \times A^{-1} = I ) (the identity matrix). Use np.linalg.inv() to find the inverse.
A = np.array([[1, 2], [3, 4]])
inverse_A = np.linalg.inv(A)
For a system of equations of the form ( Ax = B ), where ( A ) is a matrix of coefficients, ( x ) is a vector of unknowns, and ( B ) is a vector of constants, we can solve for ( x ) using np.linalg.solve().
A = np.array([[3, 1], [1, 2]])
B = np.array([9, 8])
solution = np.linalg.solve(A, B)
Eigenvalues and eigenvectors are fundamental in linear transformations, physics, and machine learning. Use np.linalg.eig() to find them.
A = np.array([[4, -2], [1, 1]])
eigenvalues, eigenvectors = np.linalg.eig(A)
The norm of a vector or matrix gives a measure of its magnitude. np.linalg.norm() calculates the norm, and you can specify the order (e.g., 1-norm, 2-norm, or infinity norm).
vector = np.array([3, 4])
matrix = np.array([[1, 2], [3, 4]])
# Euclidean norm (default is 2-norm)
norm_vector = np.linalg.norm(vector)
# Frobenius norm (for matrices)
norm_matrix = np.linalg.norm(matrix, 'fro')
The rank of a matrix represents the maximum number of linearly independent row or column vectors. np.linalg.matrix_rank() provides the rank of a matrix.
A = np.array([[1, 2], [2, 4]])
rank_A = np.linalg.matrix_rank(A)
SVD decomposes a matrix into three matrices, useful in applications like image compression and data reduction. np.linalg.svd() provides this decomposition.
A = np.array([[1, 2], [3, 4], [5, 6]])
U, S, V = np.linalg.svd(A)
| Operation | Function or Method | Description |
|---|---|---|
| Matrix Creation | np.array(), np.zeros(), np.eye() |
Create matrices and arrays |
| Addition, Subtraction | A + B, A - B |
Element-wise addition and subtraction |
| Scalar Multiplication | 2 * A |
Multiply each element by a scalar |
| Matrix Multiplication | np.dot(A, B) or A @ B |
Standard matrix multiplication |
| Transpose | A.T |
Transpose of a matrix |
| Determinant | np.linalg.det(A) |
Scalar value representing matrix characteristics |
| Inverse | np.linalg.inv(A) |
Inverse of a matrix |
| Solve Linear Equations | np.linalg.solve(A, B) |
Solves ( Ax = B ) |
| Eigenvalues and Eigenvectors | np.linalg.eig(A) |
Returns eigenvalues and eigenvectors |
| Norms | np.linalg.norm() |
Magnitude of vector/matrix |
| Rank | np.linalg.matrix_rank(A) |
Max linearly independent vectors |
| Singular Value Decomposition | np.linalg.svd(A) |
Decomposes matrix into U, S, V components |
NumPy is highly optimized for numerical calculations, making it ideal for complex linear algebra applications. Let me know if you’d like further explanations or examples for any of these functions!
Answer Key (True/False):
Watch this video for the answer:
Answer key (Mutiple Choice):
Answer Key (Fill in the Blanks):
Problem Statement: Write a Python program to access elements from a 2D list based on a specific condition. The 2D list represents a matrix of integers, and your task is to use nested loops to find and print the elements where the sum of the row index (i) and the column index (j) equals 2. These elements form a reverse diagonal in the matrix.
Input:
You are given a 3x3 matrix arr:
arr = [ [0, 1, 2], # Row 0 [3, 4, 5], # Row 1 [6, 7, 8] # Row 2 ]
Output:
Print the elements of the matrix where the sum of the row index and column index equals 2, along with their positions.
Example Output:
Element at (0,2) is 2 Element at (1,1) is 4 Element at (2,0) is 6
For more details, see Appendix A.
This section includes the data tables referred to in the text…