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Suppose that $S$ is a set of elements of $R^n$. Which of the following would prove that $S$ is a basis for $R^n$?

Select ALL that apply.

A proof that $S$ contains $n$ distinct elements.

A proof that $S$ contains $n$ linearly independent elements.

A proof that the elements of $S$ span $R^n$ and are linearly independent.

A proof that the elements of $S$ are linearly independent or span $R^n$

A proof that the elements of $S$ span $R^n$