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I'm studying engineering and for the most part, I feel like I understand the main concepts behind Linear Algebra. However, I feel like my understanding is superficial.

I feel like Linear Algebra is a subject that consists of different pieces loosely related to each other, I can't see what the most fundamental idea is behind the subject.

I would really like to know what the main idea is that connects all the different ideas in the subject.

Can someone point me in the right direction? Is it linear transformations? Is it something even deeper? I really want to know.

Linear maps, linear spaces, linear dependence, alternating forms and their interrelations.

1) The following article is from The Great Soviet Encyclopedia (1979), which sums up the mathematical relations covering subjects of classical Linear Algebra:

Linear Algebra is the part of algebra that is most important for applications. The theory of linear equations was the first problem to arise that pertained to linear algebra. The development of the theory led to the creation of the theory of determinants and subsequently to the theory of matrices and the related theories of vector spaces and linear transformations in them. Linear algebra also encompasses the theory of forms, in particular, quadratic forms, and, in part, the theory of invariants and the tensor calculus. Some branches of functional analysis constitute a further development of corresponding problems of linear algebra associated with the passage from finite-dimensional vector spaces to infinite-dimensional linear spaces.

2) A popular and explanatory article: https://betterexplained.com/articles/linear-algebra-guide/

3) There is a fascinating map on the seventh page of the book you should definitely look at https://minireference.com/static/excerpts/noBSguide2LA_preview.pdf

4) Two book recommendations

a) I.M.Gelfand - Lectures on Linear Algebra

b) S. Axler - Linear Algebra Done Right