I Like Abstract Stuff; Why Should I Care?
Dimensions of Vectors
Vectors aren't always 2-D. We can have an n-dimensional vector for any positive integer n we like. We could have a 100-dimensional vector, or a googolplex-dimensional vector.
We can also have vectors with infinitely many components.
Look at some examples.
Here's a vector with infinitely many components, all of which are 1: (1, 1, 1,...)
We could have a vector where the component in the nth place is n: (1, 2, 3, ...)
We could have a vector whose components alternate between 0 and 1: (0, 1, 0, 1, ...)
Is the last component of the vector (0, 1, 0, 1, ...) equal to 0 or a 1?
Sorry—that's a trick question. This vector keeps going forever, it doesn't have a last component.
Size of Vectors
There are quite a few different ways to measure the size of a vector v = (x1, x2, ..., xn).
This sort of measurement is called the taxicab norm or Manhattan distance. For a 2-D vector (x,y), the Manhattan distance
x + y
is the distance we need to travel from the tail of the vector to the head of the vector if we need to stay on the grid-like streets.
We've haven't defined the word norm yet. A norm is a mathematical thing that has to follow a bunch of fussy rules, but if we think of it as a way to measure size, that's close enough. A mathematician may use one norm or another norm depending on what they're doing with their vectors.
In parting, we saw that it's possible to have infinite-dimensional vectors. How could we measure the size of an infinite-dimensional vector?