(adsbygoogle = window.adsbygoogle || []).push({}); What does it mean for a vector to remain "invariant" under coordinate transformation?

I think I already know the answer to this question in a foggy, intuitive way, but I'd like a really clear explanation, if someone has it. I know all of multivariable calculus and quite a bit of linear algebra, yet I am still not sure exactly what people mean by "coordinates." My understanding is that a vector is a list of numbers. I don't know what it means for a vector to be in a certain coordinate system. It seems like the vector <2, 0, 10, 4> is just that: the ordered list of numbers two, zero, ten, four. There isn't any extra information, like "in rectangular coordinates" or "in these particular skew coordinates."

So when a vector remains invariant under a change of coordinates, or "doesn't care" about which coordinates you use, as some texts have put it, what does that mean? Does that mean the vector <2, 0, 10, 4> remains the vector <2, 0, 10, 4> or does it mean that the exact list of numbers changes, but the drawing of the vector looks the same? This seems to me like a dumb question, and I'm guessing the answer is the second option, but linear algebra is often taught so abstractly and so far removed from actual numbers that it isn't obvious to me.

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# What does it mean for a vector to remain invariant under coordinate transformation?

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