I worry that this tweet will be wildly unpopular, I do, but I must press on: Does anyone out there have a plain-English, intuitive definition of "linear transformation"?
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I've struggled with this, too. Two things which seem to help are to break it up the explanation into two pieces (scaling and addition), and in each case to talk about things which _aren't_ linear.
For instance, on the scaling: "a linear map is one where the outputs scale with the inputs. For instance, if you put twice as much in, then you get twice as much out. Double the amount you spend on oranges, and you'll double the weight of organge you'll get...
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... That sounds obvious, but there are (non-linear!) maps where that's not true. If a country doubles the amount it spends an training athletes, that doesn't mean it'll win twice as many medals."
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The same broad approach can be taken with addition. There it's very context-dependent - hopefully you have a context where some vector-like object is in the picture, and can be used to construct examples of what linearity means, and of what it does not.
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I've never ended up using these explanations. But I've occasionally pondered putting them into something introductory. In general, with abstract things like this I like the trick of explaining examples which violate the definition. It seems to help readers a lot, at least IME.
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I'll be curious to see what you come up with! I've really enjoyed many of your articles; it's fun to explain how I'd respond to this challenge. Hope it was at least a tiny bit helpful!
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