Find a basis of a subspace constructively Let $V\subset \mathbb{R}^n$ be a subspace of the standard euclidean space.  Is there a constructive method to find a basis of $V$?  

Obviously, if we can realize $V$ as the image or kernel of an explicit linear transformation, then there are methods to do this, but the only way I know to find such a linear transformation requires you already have a basis (or at least a spanning set of vectors).
If $V$ is subset of standard Euclidian space than $V$ can have dimensions 0,1,2 or 3. 
If $V$ is an euclidean plane then it has dim $2$ and the base is any of two vectors that are nonparallel and lay on that plane.
If $V$ is a line, then you can take any vector that is parallel with that line for a base.
If $V$ is a point then dim is $0$.
In any other cases you have provide additional information about $V$, for example linear operator that describes it and from there you can get vector base.
If $V$ is subset of standard Euclidian space than $V$ can have dimensions 0,1,2 or 3. 
If $V$ is an euclidean plane then it has dim $2$ and the base is any of two vectors that are nonparallel and lay on that plane.
If $V$ is a line, then you can take any vector that is parallel with that line for a base.
If $V$ is a point then dim is $0$.
In any other cases you have provide additional information about $V$, for example linear operator that describes it and from there you can get vector base.