Question

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 most one solution.
b) \(A\) is injective (hence \(n \leq k\) ). [injective means one-to-one]
c) \(\operatorname{dim} \operatorname{ker}(A)=0\).
d) \(A^{*}\) is surjective (onto).
e) The columns of \(A\) are linearly independent.