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# Foundations of Linear Algebra

(Art Institute Chicago, Chicago, Illinois - Alvin Wei-Cheng Wong)

### - Overview

Linear algebra (LA), mathematical discipline that deals with vectors and matrices and, more generally, with vector spaces and linear transformations. Unlike other parts of mathematics that are frequently invigorated by new ideas and unsolved problems, linear algebra is very well understood. Its value lies in its many applications, from mathematical physics to modern algebra and coding theory.

In the early days of AI research, linear algebra played a key role in the development of algorithms for tasks such as pattern recognition and machine learning (ML). Its ability to represent and manipulate data through vectors and matrices provides a mathematical foundation for AI pioneers to design algorithms that can learn and understand complex information.

Linear algebra has many applications, for eample:

• Calculating speed, distance, or time
• Projecting a three-dimensional view into a two-dimensional plane
• Predicting data related to decision making, medical diagnosis, statistical inferences, etc.
• Cryptography
• Computer vision
• Information security

### - Linear Algebra You Need To Know for AI

Linear algebra (LA) is the primary mathematical computing tool in artificial intelligence (AI) and many other fields of science and engineering. Using this field, you need to know about the 4 main math objects and their properties:

• Scalar, Vector , Matrix, Tensor
• Eigenvectors & Eigenvalues
• Principal component analysis (PCA)
• Singular value decomposition (SVD)
• LU decomposition
• QR decomposition/factorization
• Probability theory and statistics

Please refer to the following for more details:

### - Computational Linear Algebra

Computational linear algebra, also known as numerical linear algebra, is a subfield of numerical analysis that studies how matrix operations can be used to create computer algorithms. These algorithms can efficiently and accurately provide approximate answers to questions in continuous mathematics.

Computational linear algebra is concerned with getting computers to do matrix math with speed and with acceptable accuracy.

Linear algebra provides concepts that are crucial to many areas of computer science, including:

• Graphics, image processing, cryptography, machine learning, computer vision, optimization, graph algorithms, quantum computation, computational biology, information retrieval, web search.
• Linear algebra is also used in robotics to describe the kinematics of robots. This allows engineers to develop models that accurately describe the behavior of the system.

[More to come ...]