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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 $X=0$.
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