There's something that's always bothered me in my encounters with linear algebra, it seems like every vector they ever give is already defined according to some basis: e.g. A=<2,-1,0,5,6>.
Imagine that I have a very large circular sheet of paper so that there is no obvious x and y (or i and j). I can put a dot somewhere on the paper and draw an arrow and call it V. Then, from the same dot, I can draw another arrow and call it e1 (my first basis vector). Now, without going into details, if I just have a piece of string and a straight-edge (no rulers, protractors or other pre-made measuring devices) I can measure the length of V in terms of the length of e1 (which I take to be 1-unit long), and I can use the string to make circles and I can use the string to measure arc-lengths in terms of e1 and I can figure out the angle (in radians) between V and e1. And, with just the string and straight-edge, I can even construct e2 perpendicular to e1.
I can figure out how to do all of this in 2-dimensions without relying on a pre-existing set of basis vectors, but it requires me to actually physically draw and measure things.
My question is how do you do this in n-dimensions? Suppose I have a vector in 4-dimensional space, but that space does not yet have any defined basis. So, I define a second vector as e1. To the best of my knowledge 4D paper and 4D string do not exist in the physical world, so I have to do it abstractly. How do I determine the relative length of my two vectors and the angle between them?