Homogeneous system of equations and SVD

Question

Assume I want to solve $Ax=0$, with $A$ an $m \times n$ matrix and $x$ $n$-dimensional. Clearly this system has trivial solution $x=0$.

Which constraints in the variable $x$ are in general used in order to avoid the trivial solution? Are they application-dependent?

Suppose that I set the constraint $\|x\|_2^2=1$. In this case, I can exploit the SVD of matrix $A$. I know that all the right singular vectors $v_i$ are such that $\|v_i\|_2^2=1$. However:

$$A v_i= \sigma_i u_i $$ meaning that the choice $x=v_i$ solves the system only if $\sigma_i=0$.

Can I conclude with the following statement?

1. If $A$ is full rank, clearly the only possible solution is $x=0$ by definition, since it does not have a nullspace.
2. If I want anyway a solution in the least square sense, the best possible solution is to take $v_n$ since it is associated with the smallest singular values $sigma_n$.
3. If $A$ has rank $r< \min\{m,n\}$, then matrix $A$ has a nullspace and I can choose any $v_i$ associated with singular value $\sigma_i=0$, since they all will satisfy $A v_i=0$.

Answer 1

You are almost correct. The last columns do $\{v_i|i>rank\}$ serve as a basis for the null space. Additionally, if n>m, then your map from $\mathbb{R}^n$ to $\mathbb{R}^m$ has a nullspace(i.e., $Av_n = 0$ will hold). The least-squares argument holds otherwise.