I am currently following a long this video, which is an introduction to Machine Learning. (https://www.youtube.com/watch?v=esTIhqAFKu4&list=PLGd9Gn0_Oc65os52iy4jwvex2q50r5q0U&index=27&t=2818s)
At 47:15 to 48:45 the professor goes on to talk about, how "non normalizing" (sorry, I have a hard time hearing him even with subtitles on its a mess) the dimension of the vector goes down by 1. So having a vector in the realm of R^(p) gives a dimension of R^(p-1). He even uses the example of if a vector was in R^2 than it would be R^1. This seems wrong, but it might be me who have a hard time understanding him. Can someone brighter than me shine some lights on this? Thank you very much.
The lecturer is saying "norm normalized", where "norm" refers to the usual Euclidean length/norm function written $|| \cdot ||$. If $x \in \mathbb{R}^p$ is nonzero then if we "norm normalize" it the result is $x / ||x||$. That's got norm 1, so it belongs to the $(p-1)$ sphere $S^{p-1}$ (because this is defined to be the set of vectors in $\mathbb{R}^p$ with norm 1).
For example, if you take a nonzero vector $x \in \mathbb{R}^2$ then normalize it you get something on the unit circle $S^1$.