Prove Au and Av Linearly Independent

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Homework Statement


Let u_1...u_n be linearly independent column vectors in R^n and A an invertible n x n matrix. Prove that the vectors Au_1...Au_n are linearly independent.


Homework Equations





The Attempt at a Solution



It is easy to prove this using scalars and the definition of linear independence. But, then why is this relevant to invertible matrices? Is there a way to prove this using column spaces, row spaces, null spaces, etc?
 
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It is not true if A is not invertible!
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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