Background: I'm trying to show that the transformation $T:\Bbb R^n\to\Bbb R^n$ defined by $T(x_1,\dots,x_n) := (|x_2-x_1|,|x_3-x_2|,\dots,|x_1-x_n|)$ is (or is not, this is out of curiosity only) bijective (or if not, bijective up to sign? injective only? surjective only?). The problem reduces to two highly elementary systems of absolute value problems as follows.
Injectivity: Given that $$|x_2-x_1| = |y_2-y_1|\\ |x_3-x_2| = |y_3-y_2|\\ \dots\\ |x_1-x_n| = |y_1-y_n|$$ is it necessarily true that $x_1=y_1,\dots,x_n=y_n$?
Surjectivity: Given a sequence $c_1,\dots,c_n$ of numbers, can we necessarily find another sequence $x_1,\dots,x_n$ such that $$|x_2-x_1| = c_1\\ |x_3-x_2| = c_2\\ \dots\\ |x_1-x_n| = c_n$$
Taking an initial look at these somewhat similar problems, my immediate gut feeling is that surjectivity is somewhat more obvious than injectivity. I've never, however, done anything quite like these, can someone point me in the right direction?