We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.
Points, Vectors, and Functions

Points, Vectors, and Functions

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, ...)

Sample Problem

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 = (x1x2, ..., 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?

People who Shmooped this also Shmooped...