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=[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?
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?