Fundamental problem of Linear Algebra

Question

What is the fundamental problem of linear algebra? I understand it is a big question and not easy to explain completely, and seems no way to prove an answer is correct. I just wanna listen to you experts' opinion. For example, can I say that to solve linear equation systems is fundamental in linear algebra? Can I say that matrices is the fundamental study object of linear algebra?

Answer 1

I'd say that the fundamental problem of linear algebra is to study linear transformations. You don't get very far without assuming something extra.

Finite dimension is the first important assumption. In this context, the main result is probably the rank-nullity theorem and its matrix equivalent, the existence of LU decomposition.

That the scalar field is $\mathbb R$ or $\mathbb C$ is the second important assumption. Then the main results are:

• For a general linear transformation between different spaces, the major theorem is the singular value decomposition.

• Much can be said about operators in a space, that is, a linear transformation from one space to itself. SVD still applies, but the main concept here is invariant subspaces, so that the operator (and hence its matrices) can be expressed in terms of simpler ones. A typical major result is the existence of unitary diagonalization of normal operators. This is the spectral theorem.

Answer 2

Your question is not entirely clear and I am not sure what exactly you mean by fundamental, but it seems you're looking for an answer to one or some of the following questions:

• What is it that one studies in linear algebra? That is, what are the "fundamental" objects with which linear algebra is concerned?
• What problem initially necessitated the invention of linear algebra as a tool for its solution?
• What problems are "linear algebra problems"?

You seem to have generally correct, if limited, notion about the first two. That is a large part of linear algebra is indeed the study of matrices, and the initial reason for linear algebra's invention was indeed the solution of linear systems of equations.

However, there is more to linear algebra. Linear algebra can be more accurately described as the study of vector spaces (usually of finite and countably infinite dimension) and the linear mappings between them. Matrices are conceptualized not just as box with numbers in them, but as a shorthand for some linear mapping whose composition and application may be carried out via "matrix-multiplication."

It so happens that matrix multiplication allows for some very useful manipulations and representations of these boxes with numbers, which is why linear algebra has been so incredibly useful across superficially unrelated fields of mathematics. In many instances of its application, linear algebra can be thought as either an organizational tool (e.g. in representation theory) and a useful approximate model (e.g. in calculus).

I don't know if any one problem is fundamental to linear algebra. The question of whether a linear system can be solved certainly got things started, but linear algebra has blossomed from there. A lot of problems come down to classifying "which matrices are nice" or "finding a convenient way to write a matrix". See, for example, the problem of diagonalizability, normal/Hermitian/positive definite matrices, singular value decomposition, and L-U factorization.