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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 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 $\mathbb{R}^{k}$.
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