Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map. Show that the following are equivalent.

a) For every \(y \in \mathbb{R}^{k}\) the equation \(A x=y\) has at least one solution.

b) \(A\) is surjective (hence \(n \geq k\) ). [surjective means onto]

c) \(\operatorname{dim} \operatorname{im}(A)=k\).

d) \(A^{*}\) is injective (one-to-one).

e) The columns of \(A\) span \(\mathbb{R}^{k}\).