Given any $k$-tupel of linearly independent vectors $X$ as above, there exists a $k$-tuple $Y$ biorthognal to it. If $k=n$, this $Y$ is unique.

解答: Since $$\bex \rank(X^*X)=\rank(X)=k, \eex$$ there exists an unique $A\in M_k$ such that $$\bex X^*XA=I_k. \eex$$ Take $Y=XA$, we are completed. 

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.1.1