Say you have $k$ linear algebraic equations in $n$ variables; in matrix form we write $AX=Y$. Give a proof or a 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 $AX=Y$ .
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 $AX=Y$ .
e) If $n < k$ the only solution of $AX=0$ is $X=0$ .
I solved c), answer:
Rank Theorem: $\dim N(A) = n − \dim I(A) \geq \dim I(A) \geq n-k \geq 1 $.
For a) I think the zero matrix is a valid counterexample, but I don't know how to argue it.
For b) when $A$ is the zero matrix. Is it mean that zeros elements matrix for A? What about $X$ and $Y$?
e) Let $A$ be the zero matrix. Can anyone give a counterexample for that please?