(1) Can work with any algebraic field. The greatest interest in practice is for the field the set of real numbers R. The set of complex numbers C can also be important. Curiously, there are some applications, say, error correcting codes, where the field is finite, e.g., the set of integers modulo a prime number.

For learning, I recommend working with the complex numbers C for where they do help, with very little extra effort, and let the real case be mostly a special case.

(2) The main motivation for linear algebra is just solving systems of simultaneous linear equations as in high school. The main solution technique is just the elementary row operations used as in Gauss elimination -- can learn that in three minutes and program it in 10.

(3) Matrix theory is central, even crucial, in linear algebra but really is just some darned convenient notation that can motivate just from solving a system of linear equations. So, for positive integers m and n, an m x n matrix is a rectangular array of numbers (real, complex, other, as you have decided to work with) with m rows and n columns. So, it's just notation.

(4) The big deal about matrix theory is just that can multiply matrices. The multiplication is defined just so that a system of linear equations can be written as a product of (A) a matrix from the coefficients and (B)+ a matrix with one column of the unknowns set equal to a matrix of one column for the right hand side. Let's try it for some high school case of 2 equations in two unknowns (roughly as in TeX, and underscore character starts a subscript):

a_11 x_1 _ a_12 x_2 = b_1

a_21 x_1 _ a_22 x_2 = b_2

So, the 2 unknowns are x_1 and x_2.

/ a_11 a_12 \
2 x 2 A = | |
\ a_21 a_22 /

/ x_1 \
2 x 1 x = | x_2 |
\ x_3 /

and similarly for 1 x 1 b. Then the matrix multiplication Ax is defined just so that we can write Ax = b and have that be equivalent to the 2 equations in 2 unknowns in the high school form above.

Commonly we write A = [a_ij] for i = 1, 2, ..., m and j = 1, 2, ..., n.

Now, we can do some arithmetic on the matrices, and that starts to get darned useful: E.g., for a number p we can define the product pA = [p a_ij]. So we multiply the matix A by the number (call it a scalar if you want) p. And we can add two matrices: So for m x n A = [a_ij] and m x b B = [b_ij] we can define m x n C = A + B where C = [a_ij + b_ij]. So just add the corresponding components (elements, entries).

Then presto, bingo, we get to why this subject is called linear algebra: Given m x n A, n x 1 x, n x 1 y, and numbers p and q, we have

A(px + qy) = (pAx) + (qAy)

which says that A is linear or a linear operator -- in calculus, both differentiation and integration are also linear operators. There is a huge book Linear Operators by Dunford and Schwartz, that is, functional analysis, that is where the points we are talking about, analyzing, actually functions. So, biggie special cases are Hilbert space and Banach space. So, linearity in linear algebra is a biggie start on functional analysis.

Why care about linearity? Because its (A) simple, (B) permits powerful theorems fairly easy to prove, and (C) is an assumption that often holds quite well in applications in science and engineering.

The next biggie deal about matrix multiplication is that it's associative:

(AB)C = A(BC)

and that property drives a lot in duality, game theory, matrix computations, error correcting coding theory, etc. Can write out a proof in terms of the i's and j's and, there, will see that the crucial step is being able to interchange the order of summation signs (which in calculus is the same as interchanging the order of integral signs which in more advanced parts of calculus is Fubini's theorem) when adding finitely many numbers but could also know that the result is true because A, B, and C are functions, function composition is essentially just matrix multiplication, and function composition is essentially obviously associative!

Given m x n matrix A with m = n, then A is square. If a_ij = a_ji, that is, if reflect about the diagonal from upper left to lower right and get the same thing, the A is symmetric.

Now a biggie, IMHO the most important result in linear algebra: Regard