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This not a homework question. I'm reviewing some linear algebra and I found this on a worksheet. I just need a hint on how to approach this problem.

Let β=[itex]\{ v_1,v_2,...,v_n\}[/itex] be a basis for [itex]R^n [/itex]. Let M be the matrix whose columns are the basis vectors in β. Do the columns of [itex] M^2 [/itex] form a basis?

I've played around with some examples and it seems to be true. I know that M has full rank and hence invertible, therefore, [itex] M^2 [/itex] is also invertible and has full rank. But this only means the columns of [itex] M^2 [/itex] spans the vector space. How do I show the columns are also linearly independent?

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# Columns of Matrix Form a Basis?

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