Proving X is Full Column Rank Matrix if X^TX is Non-Singular

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SUMMARY

X is a full column rank matrix if and only if X^TX is non-singular (invertible). This conclusion is established through the properties of matrix rank and the implications of invertibility. Specifically, if X has full column rank, then the columns of X are linearly independent, leading to a non-zero determinant for X^TX. Conversely, if X^TX is non-singular, it confirms that the columns of X cannot be linearly dependent, thus ensuring full column rank.

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Homework Statement


Show (in complete detail) that X is a full column rank matrix if and only if
X^TX is non-singular (invertible). Assume X is a real matrix.

X^T is X transpose

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The Attempt at a Solution

 
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What does the fact that X is of full rank imply about X? Secondly what does the fact that (X^TX) being invertible imply about what the previous statement concludes?
 


Show an attempt at solving the problem, please? Or at least say why you can't.
 

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