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Let $A$ be an $n \times n$ matrix of real or complex numbers. Which of the following statements are equivalent to: "the matrix $A$ is invertible"?

a) The columns of $A$ are linearly independent.
b) The columns of $A \operatorname{span} \mathbb{R}^{n}$.
c) The rows of $A$ are linearly independent.
d) The kernel of $A$ is 0 .
e) The only solution of the homogeneous equations $A x=0$ is $x=0$.
f) The linear transformation $T_{A}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ defined by $A$ is 1-1.
g) The linear transformation $T_{A}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ defined by $A$ is onto.
h) The rank of $A$ is $n$.
i) The adjoint, $A^{*}$, is invertible.
j) $\operatorname{det} A \neq 0$.
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