Question: Gaussian elimination is a useful tool figure out whether a set of vectors spans a vector space and if they are linearly independent. Consider a matrix \(M\) made from an ordered set of column vectors \(\left(v_{1}, v_{2}, \ldots, v_{m}\right) \subset\) \(\mathbb{R}^{n}\) and the three cases listed below:

1. \(\operatorname{RREF}(M)\) is the identity matrix.
2. \(\operatorname{RREF}(M)\) has a row of zeros.
3. Neither case i or ii apply.

First give an explicit example for each case, state whether the column vectors you use are linearly independent or spanning in each case. Then, in general, determine whether \(\left(v_{1}, v_{2}, \ldots, v_{m}\right)\) are linearly independent and/or spanning \(\mathbb{R}^{n}\) in each of the three cases. If they are linearly dependent, does \(\operatorname{RREF}(M)\) tell you which vectors could be removed to yield an independent set of vectors?

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