All the linear algebra you need before entering the AI realm.

Why do I need linear algebra when I have prebuilt AI libraries?

Understanding Models:

AI models, particularly those involving deep learning, are built upon linear algebra concepts. Understanding the underlying mathematics helps you comprehend how these models work, what their limitations are, and how to troubleshoot or improve them.

Customization and Optimization:

Prebuilt libraries offer a lot of functionalities, but there are scenarios where you might need to customize or optimize the models for your specific use case. This often requires a deep understanding of the linear algebra that these models are based on.

Efficiency:

Knowing linear algebra can help you write more efficient code. For example, you can optimize matrix operations, reduce computational complexity, and make your algorithms more scalable.

1. Vectors and Vector Spaces

Vectors: are fundamental to representing data in AI. Each data point can be represented as a vector in a high-dimensional space.

Vector spaces: provide the framework for operations involving vectors, such as addition and scalar multiplication, which are used extensively in algorithms and data manipulation.

2. Matrices and Matrix Operations

Matrices: are used to represent and manipulate large datasets, perform transformations, and manage parameters in neural networks.

Matrix operations: like addition, multiplication, and inversion are crucial for implementing various algorithms, including those used in machine learning models and neural networks.

3. Matrix Decompositions

Eigenvalue decomposition and singular value decomposition (SVD) are used in techniques like Principal Component Analysis (PCA) for dimensionality reduction, which helps in preprocessing data to make it more manageable and meaningful.

Understanding these decompositions aids in optimizing and simplifying complex computations.

4. Linear Transformations

Linear transformations help in understanding how data points are mapped from one space to another, which is fundamental in algorithms for data preprocessing and feature extraction.

They also form the basis of operations in neural networks where inputs are transformed through layers.

5. Systems of Linear Equations

Solving systems of linear equations is essential for optimization problems in AI, where the objective is to minimize or maximize a particular function, such as in linear regression or support vector machines (SVMs).

It also helps in understanding the behavior and constraints of models.

6. Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are critical in understanding the properties of matrices used in algorithms, such as stability and behavior under transformation.

They are used in PCA and other algorithms to identify the principal components and reduce the dimensionality of data.

7. Norms and Distance Measures

Norms are used to measure the size or length of vectors, which is important for evaluating distances and similarities between data points.

Understanding different distance measures is crucial for clustering algorithms, nearest neighbor searches, and other similarity-based techniques.

8. Orthogonality and Orthonormal Bases

Orthogonality simplifies many mathematical computations and is used in algorithms for simplifying data representations.

Orthonormal bases are used in Fourier transforms and other methods for signal processing and feature extraction.

9. Projections

Projections are used to reduce the dimensionality of data and to project data points onto subspaces, which is fundamental in many machine learning algorithms, including PCA and linear regression.

10. From Theory to Practice: Implementing Linear Algebra in Code

Go from theoretical understanding to hands-on implementation as we delve into programming with linear algebra. Learn how to translate mathematical concepts into efficient and effective code with detailed examples and applications.

Resources To Learn Linear Algebra

Mathematics for Machine Learning: Linear Algebra

Linear Algebra and Its Applications by David C. Lay

Matrix Computations by Gene H. Golub and Charles F. Van Loan

3Blue1Brown - Essence of Linear Algebra

NumPy Documentation

SciPy Documentation

SymPy Documentation