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Numerical linear algebra is the intersection of numerical analysis and linear algebra: its purpose is the design and analysis of algorithms for the numerical solution of matrix problems. The most important problems are the solution of a system of linear equations and the determination of eigenvalues.

Linear algebra (LA) plays a fundamental role in the development of algorithms for tasks including machine learning, pattern recognition, etc. It fuels Artificial Intelligence (AI) by providing essential concepts and mathematical tools for representing, manipulating, transforming, optimizing, and learning data.

Functional analysis studies function spaces. These are vector spaces with additional structure, such as Hilbert spaces. Linear algebra is thus a fundamental part of functional analysis and its applications, which include, in particular, quantum mechanics (wave functions) and Fourier analysis (orthogonal basis).

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 and web search.

Linear algebra is the math of vectors and matrices. In statistics, the main purpose of linear algebra is to organize data and write down the manipulations we want to do to them. A vector of length n is also called an n-tuple, or an ordered sequence of length n. We can suppose that each data point is a real number.

Not really, at least for college level introductory courses. You would need the basics of algebra ( manipulation of equations ) to do real analysis. More advanced courses like functional analysis might need a good helping of linear algebra, but that's not what we call real analysis.

How is linear algebra used in data science? Loss functions, regularization, support vector classification, image recognition, dimensionality reduction and many other machine learning techniques are all applications of linear algebra.