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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.
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