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I don't think Sheldon Axler wants to literally banish determinants, they are a useful tool. But they do obscure the meaning in a lot of proofs. If you start us
by sgdpk 4y ago
I don't think Sheldon Axler wants to literally banish determinants, they are a useful tool.
But they do obscure the meaning in a lot of proofs. If you start using them immediately, it is also hard to motivate where they come from, what's the intuition behind them and how the students could arrive at the definition themselves.
And even if you use them and they produce short one-line proofs, you should also prove all the properties behind them first, which is where all the complexity is hiding.
The nice point about Linear Algebra Done Right is that it explains how Linear Algebra works without resorting to "determinant magic". In the end, you will also understand why determinants work so well in so many proofs.
In any case, his book is really suited for a second course in Linear Algebra, so maybe it is beneficial to start using determinants right away. Personally I understood them better with his book.
Disclaimer: I don't know Sheldon Axler, but I read his book.
- zmmmmm 4y agothis is definitely one of the problems I have always had. There seems to be so much "lore" hidden in various axioms about these things that often some derived concept like determinants that seems to relate to nothing intuitive are taught to the exclusion of insight generating concepts. They seem to think it's a clever way to teach to do this and then later on reveal the magic behind the curtains, saying "see this is how determinants work!". But for me I was lost and gave up trying to understand 6 lectures ago.