In the book "First Course in Linear Algebra" by Bhattacharya-Jain-Naugpal, there is a different view for the development of Jordan basis.
I find it difficult the proof of the Theorem 3.3 in the book, p. 184. I will be happy if anyone provides motivation for the way followed by authors and also for the proof of Theorem 3.2 and 3.3 in above link. When I presented the proof in classroom without motivation (which I didn't found myself), many students didn't attend my class until the end of the theory of Jordan canonical form. Can anybody help me for motivation?