Why do we impose a constraint on a Homogeneous Least Square Problem?

Question

The homogeneous least square problem given by : $$ \mathbf{Ax}=\mathbf{0} $$ admits a trivial solution assuming full rank for $\mathbf{A}$ and this trivial solution is $$ \mathbf{x=0} $$ However, it is regularly not the case and a constraint is usually imposed on $\mathbf{x}$ $$ \|\mathbf{x}\|_{2}=1 $$ My questions are :

1. What's a purpose to have a constraint of $\|x\|_{2}=1$ other than trying to get a solution that isn't trivial?

2. Why is the constraint $\|x\|_{2}=1$? why not $\|x\|_{2}=2$?

Answer 1

1. The trivial solution is both uninteresting and for most applications, not useful. Since homogeneous linear equations always admit a zero solution, regardless of the matrix, such solutions tell nothing about the problem. And usually, whatever purpose you have in solving this linear equation is not going to be served by a zero solution. For example, it won't tell you where to head to maximize your machine learning. It says "don't go anywhere, this spot is good enough".
2. If $\mathbf x$ is a solution to $\mathbf {Ax} = \mathbf 0$, then so is $k\mathbf x$ for any scalar $k$. And $\|k\mathbf x\| = |k|\|\mathbf x\|$. So if you have any non-zero solution $\bf x$ to an homogeous linear equation, then $\hat {\bf x} = \frac 1{\|\bf x\|}\bf x$ is also a solution and satisfies $\|\hat {\bf x} \| = 1$. In later calculations, unit vectors are generally easier to deal with. You don't have to keep dividing things by their norm. So if you can have solutions of any non-zero size, why not just stick with the size that is most convenient?