Do Columns of M² Form a Basis?

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SUMMARY

The discussion centers on whether the columns of the matrix M², formed from a basis β = {v₁, v₂, ..., vₙ} for Rⁿ, also constitute a basis. It is established that since M has full rank and is invertible, M² retains these properties. However, the key question remains whether the columns of M² are linearly independent, which is essential for them to form a basis. The conclusion drawn is that while M² spans the vector space, further proof is required to confirm the linear independence of its columns.

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  • Understanding of linear algebra concepts, specifically basis and rank.
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puhsyers
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Hi everyone,

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=\{ v_1,v_2,...,v_n\} be a basis for R^n. Let M be the matrix whose columns are the basis vectors in β. Do the columns of M^2 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, M^2 is also invertible and has full rank. But this only means the columns of M^2 spans the vector space. How do I show the columns are also linearly independent?
 
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If the columns are linearly dependent, then it doesn't have full rank. My definition of rank is the dimension of the space spanned by the columns.
 

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