Well, thing is, Linear Algebra isn't matrix manipulation. Matrices are just one aspect of Linear Algebra - it's like saying that symbols such as 'x', 'y' and 'z' are "Algebra."

Matrices are representations of (linear) transformations between Linear Spaces. Linear Algebra is talking about the Linear Spaces, and manipulating matrices is sort-of like finding roots of or factoring polynomials.

Which brings me to your second part. Vectors are elements of vector spaces. Matrices are transformations between vector spaces. The one is not the other, and while there are similarities and overlap (and sometimes identifying aspects of each to the other is useful), conflating the two at an early stage is a recipe for getting no further than simple matrix manipulation. If that's all you want then that's OK, but you miss out on a lot of insights that can be really valuable and make you stand out from the crowd.

But remember, I'm saying this as a mathematician, and my terminology will be different. I mostly want to make people aware that they are using terms in a different way, and open the opportunity for the kind of confusion I've suffered for decades.

For the programmers out there, it's a bit like the difference between writing code which takes concrete data types ('n-tuple of floats', 'n by m matrix of floats'), and writing more generic code which takes abstract datatypes (roughly: 'thing supporting addition and scalar multiplication' and 'function on said things which happens to satisfy a linearity property').

The abstraction brings some benefits, but also has some conceptual overhead for those new to the area.

In a sense the abstraction isn't technically necessary, at least for finite-dimensional vector spaces: every finite-dimensional real vector space is isomorphic to some R^n, so you can express all the results of finite-dimensional linear algebra in purely concrete terms (vectors, matrices) if you really want to. In particular, if you're only interested in doing geometry in R^2 and R^3, the abstract vector space treatment adds relatively little.

But (again perhaps analogously to programming) hiding away the lower-level details can make things clearer and more elegant if one invests some time in getting comfortable with the abstractions involved. It allows you to use whichever representation comes most naturally for your vectors, rather than tying you to representing in terms of some particular basis. And it does add some generality, especially when it comes to infinite-dimensional spaces, which turn out to be pretty useful things.

> ", thing is, Linear Algebra isn't matrix manipulation. Matrices are just one aspect of Linear Algebra - it's like saying that symbols such as 'x', 'y' and 'z' are 'Algebra.'"

Well, I see your point, but to many, x, y and z are algebra. When public schools teach solving strategies like substitution, polynomial factoring, or concepts like linear equations, series, and exponents, these are labeled "algebra" and are basic extensions of solving for some variable(s).

So it seems you are arguing that the public understanding of "Algebra" and a mathematician's understanding of "Algebra" are different. (And the same for "linear algebra".) ... because the public's understanding is based largely on a shared experience in public schools.

I'd be surprised if it weren't so, and I suspect the problem of a different interpretation of terms is the case in many fields.

To some people, that big TV with pictures in it is called a computer. To you, it's probably a monitor. Are they just as right as you are?