The first thing the textbook does is draw an arrow. The second thing it does is write a column of numbers. The third thing — eventually — is to talk about a “vector space” as if you should already know what that is.
I spent more time than I want to admit, in undergrad, treating these three things as the same idea worded three ways. They are not the same idea. They are three different framings of an object, each useful at a different point in the curriculum, and the textbook doesn’t tell you which framing it is currently using.
The arrow
The arrow framing is geometric. It has a tail, a head, a length, and a direction. You add two arrows by sliding the second one’s tail to the first one’s head. The thing you get is an arrow.
This framing is for intuition. It is excellent for two and three dimensions and quietly useless past that. You cannot draw a seven-dimensional arrow. You can pretend to, but the picture is doing nothing for you.
The list of numbers
The list framing is what gets typed into a computer. A vector is [3, -1, 2]. You add two vectors by adding their entries, in order. You scale a vector by multiplying every entry by the same number.
This framing is for computation. It is what numpy actually does. It survives going to seven dimensions, or seven hundred, without complaint. It is also where students start to lose the geometric thread, because nothing about a list of numbers feels like an arrow.
The abstract object
The abstract framing is what the upper-division courses care about. A vector is anything that lives in a vector space — meaning, anything you can add to other vectors and scale by numbers, where the additions and scalings obey a small list of rules.
The reason this framing exists is that the same set of rules turns out to apply to a lot of things that aren’t arrows and aren’t lists. Once you’ve internalized that, an enormous amount of the rest of linear algebra becomes a lot less mysterious.
Why this matters for what comes next
The next post is about matrices, and matrices are the place where the three framings have to coexist on the page at the same time. A matrix is a list of numbers (computational), it transforms one arrow into another arrow (geometric), and it is itself a function between vector spaces (abstract).
If you don’t have all three vector framings loaded simultaneously, the matrix conversation gets harder than it needs to be. So this is the warm-up.