In the decomposition a = aTT + aNN, what do aT and aN represent?

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 10EQ
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In the decomposition a = aTT + aNN, what do aT and aN represent? 

 
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