Suppose that u is a d x 1 vector such that ||u|| =1. What information does it implied to us?

Question

We know that the the norm of a vector is $\| \vec{u} \| = \sqrt{u_1^2 + u_2^2 + ... + u_n^2}$, where $\vec{u} \in \mathbb{R}^n$. How does the d x 1 vector look like using the definition of norm, and what does $\mathbb{R}^n$ really means? Or what is $\mathbb{R}^n$ for the d x 1 vector?

Answer 1

It is on the unit sphere.

It might help to work with lower specific number of $n$, say $n=1,2,3$.

If $n=1$, then $u=1$ or $u=-1$.

If $n=2$, then it lies on the unit circle, $u_1^2+u_2^2=1$.

If $n=3$, then it lies on the unit circle, $u_1^2+u_2^2+u_3^2=1$

$\mathbb{R}^2$ means $\{(x,y)^T|x, y \in \mathbb{R}\}$, similary for general $\mathbb{R}^d$.