Question

Say you have \(k\) linear algebraic equations in \(n\) variables; in matrix form we write \(A X=Y\). Give a proof or counterexample for each of the following.

a) If \(n=k\) there is always at most one solution.
b) If \(n>k\) you can always solve \(A X=Y\).
c) If \(n>k\) the nullspace of \(A\) has dimension greater than zero.
d) If \(n<k\) then for some \(Y\) there is no solution of \(A X=Y\).
e) If \(n<k\) the only solution of \(A X=0\) is \(X=0\).