The fundamental idea behind Linear Algebra 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. 
 A: 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
A: Linear maps, linear spaces, linear dependence, alternating forms and their interrelations.
A: I struggled through the same thing in engineering school.
Part of the issue is that it is easy to see how linear algebra works in 2- or 3-dimensional Euclidean space.  It's the most basic algebra, solving simultaneous equations with straight lines.  In fact, most of what you study in classic high school algebra is about getting away from such a simplistic and rigid constraint - i.e., linearity in $x$ and $y$.  So, I agree, a lot of the formalism you get in engineering school feels almost like clever notation to explain the most basic algebra in n-dimensional space.  Notably, matrix algebra feels like a very compact notational scheme for solving simultaneous equations.
There are a few things that will become more clear over time, however, as you study linear algebra and other applications of it, and they come from greater generalization. What you find is that "linear algebra" (with a little 'l', meaning Euclidean space) is really a degenerate case of a much broader set of principles and tools. 
For example, eventually you learn in solving differential equations that a great many ones of practical interest have the form
$$y(t)=\sum_n \sin(n\omega t)$$
where $n$ may be infinity.  Now, it will turn out that $\sin(n\omega t)$ for a set of $n$'s can be thought of as the (orthogonal) basis for a linear space, just as $\hat i,\hat j, \hat k$ form the basis for a linear Euclidean space with axes $x, y,$ and $z$.  And a whole bunch of the machinery that applies to Euclidean space will work here as well.  
