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# AI and Linear Algebra

(University of Michigan at Ann Arbor)

### - Overview

Linear algebra (LA) is the mathematical foundation at the heart of artificial intelligence (AI), which has sparked profound changes in countless aspects of our existence, heralded an era of industrial transformation, and defined the trajectory of technological progress.

Linear Algebra is the mathematical foundation that solves the problem of representing data as well as computations in machine learning (ML) models. It is the math of arrays - technically referred to as vectors, matrices, and tensors.

Linear Algebra is the backbone of ML model algorithm in Data Science. By understanding the historical background and basic concepts of linear algebra, we can grasp its important role in ML, reasoning, and decision-making.

Linear algebra is the branch of mathematics concerning linear equations and linear functions and their representations through matrices and vector spaces. Linear algebra is central to almost all areas of mathematics.

Linear algebra is the study of linear sets of equations and their transformation properties. Linear algebra allows the analysis of rotations in space, least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering.

Confusingly, linear algebra is not actually an algebra in the technical sense of the word "algebra" (i.e., a vector space  over a field , and so on).

### - Four Pillars of Linear Algebra

Linear algebra is a field of mathematics that is universally agreed to be a prerequisite to a deeper understanding of ML. Although linear algebra is a large field with many esoteric theories and findings, the nuts and bolts tools and notations taken from the field are practical for ML practitioners.

Linear algebra is the mathematics of data. It has had a marked impact on the field of statistics. Linear algebra underlies many practical mathematical tools, such as Fourier series and computer graphics.

The four pillars of linear algebra are vectors, matrices, tensors, and scalars. Linear algebra is a branch of mathematics that is considered a computational tool for science, engineering, and data analytics. It deals with vector spaces, linear transformations, and systems of linear equations.

Here are some other key concepts in linear algebra:

• Vector spaces: A fundamental concept for linear algebra, along with matrices, which allow for computation in vector spaces. This provides a way to study and manipulate systems of linear equations.
• Linear transformations: A central part of linear algebra, including scaling, rotation, and inversion. Scaling a vector stretches or squeezes it, rotation turns the vector space, and inversion flips the vector space.
• Fundamental linear subspaces: The four fundamental linear subspaces of an m×n matrix A are the right and left null spaces, the column space, and the row space.

### - Linear Algebra Fuels AI and ML

The fuel of ML models, that is data, needs to be converted into arrays before you can feed it into your models. The computations performed on these arrays include operations like matrix multiplication (dot product). This further returns the output that is also represented as a transformed matrix/tensor of numbers.

Linear algebra is important for AI and ML because it helps understand and manipulate data. Concepts such as vectors, matrices, matrix multiplication and eigenvectors are fundamental concepts. Linear algebra can be used for data clustering, classification, validation, and fitting.

Linear algebra is also important for understanding the theory behind ML, especially deep learning. It helps you make better decisions by giving you a better visual understanding of how the algorithm works.

The principle of AI is that human intelligence can be defined in ways that machines can imitate. However, applied mathematics can be processed by machines, but theoretical mathematics cannot.

### - Linear Algebra: A Key Component of AI Algorithms

There is a profound relationship between linear algebra and AI. Linear algebra plays an important role in the development of algorithms and technologies that support AI applications.

Linear algebra is a key component of AI algorithms. It's used to handle and analyze large amounts of data, especially unstructured data like speech, text, and images.

Linear algebra is also used as a "data guru" for ML and AI. It's used for data fitting, classification, validation, and clustering.

Linear algebra is also important in data science. It's the foundation of ML algorithms, and it enables operations like matrix multiplication, which are essential for model training and prediction.

One of the most important applications of linear algebra in AI is data fitting. Data fitting is the process of creating a mathematical function or curve fitting that best fits a set of data points.

### - Linear Algebra in Machine Learning

Linear algebra is a mathematical foundation that is used in machine learning (ML) to solve data problems and represent data. It is considered a pillar of ML, and many recommend it as a prerequisite subject for ML.

Here are some ways linear algebra is used in ML:

• ML algorithms: Linear algebra is the backbone of ML algorithms, enabling operations like matrix multiplication.
• Data representation: Linear algebra is the math of arrays, which are technically referred to as vectors, matrices, and tensors.
• Neural networks: Linear algebra operations describe the transformations of information as it flows from one layer to another.
• Feature extraction techniques: Matrices are used in feature extraction techniques such as Principal Component Analysis (PCA) and Singular Value Decomposition (SVD). These methods transform high-dimensional data into a lower-dimensional space using matrix operations.

Linear algebra can also help develop the intuition and awareness required for success in machine learning and data science.

### - Linear Algebra in Neural Networks

Linear algebra is a fundamental part of many AI applications, including neural networks. Neural networks are mathematical models that combine biology, statistics, and linear algebra to solve problems.

Linear algebra is used to:

• Analyze and understand the properties of operations and how they affect the network's behavior
• Perform a Principal Component Analysis to reduce the dimensionality of data
• Represent and process networks

Linear algebra operations describe the transformations of information as it flows from one layer to another.

A neural network is a mathematical model that takes inputs and calculates outputs to target the actual result. A linear neural network is a neural network that only uses linear transformations in its layers, such as matrix multiplication and addition.

Some popular types of deep neural networks include:

• Multi-Layer Perceptrons (MLP)
• Convolutional Neural Networks (CNN)
• Recurrent Neural Networks (RNN)

### - Algebraic Machine Learning

Algebraic machine learning (AML) is a new AI paradigm that combines user-defined symbols with self-generated symbols, allowing AML to learn from data and adapt to the world like a neural network, combined with the interpretability capabilities of symbolic AI .

AML is a purely symbolic method that neither uses neurons nor is it a neurosymbolic method. Algebraic machine learning does not use parameters and does not rely on fitting, regression, backtracking, constraint satisfiability, logical rules, production rules, or error minimization.

AML has shown accuracy in challenging tasks like the MNIST dataset and N-Queens completion, but its use of two semi-lattices can be computationally demanding.

### - Linear Algebra Topics

The linear algebra prerequisite should include the following topics:

• Mathematical operations with matrices (addition, multiplication)
• Matrix inverses and determinants
• Solving systems of equations with matrices
• Euclidean vector spaces
• Eigenvalues and eigenvectors
• Orthogonal matrices
• Positive definite matrices
• Linear transformations
• Projections
• Linear dependence and independence
• Singular value decomposition

[More to come ...]