Posts tagged with tangent

The dual V* of a vector space V  over ℝ matches lists of reals to linear functionals.

What’s the simplest way to say this? Talk about a number like “5”. Initially I think of it as 5 stones ⬤⬤⬤⬤⬤. But I could also imagine a line through the origin with a slope of 5, representing the verb quintuple.

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linear maps as multiplication
linear mappings -- notice they're ALL straight lines through the origin!

pictures of lines through the origin with various slopes

Seen as a function ƒ₅=quintuple, the-line-through-the-origin-with-a-slope-of-5, is ƒ₅(x)=5•x. That ƒ₅ does things like

  • ƒ₅(■■■)=5+5+5 and
  • ƒ₅(■■■■■■)=5+5+5+5+5+5.

Counting in the dual space ƒ₀,ƒ₁,ƒ₂,… would look like _ / ∕ ...|. Increasing slope from _ to ⁄ to | instead of increasing number from 0 to 1 to ∞. Or I could say id, double, triple, quadruple, quintuple, ….

(Why did I jump so suddenly 0,1,… from _ flat to ⁄  45°? This just proves that half of the ℝ⁺ are stuffed between [0,1) and the other half are between (1,∞).
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To jump between the two worlds you use the reciprocal map flip(■)≝1/■. T
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hen you’d be counting id, half, third, fourth, fifth, sixth, seventh… Infinity in a teacup.)

 

These two things—the five rocks ⬤⬤⬤⬤⬤ and the function ƒ₅—aren’t even the same kind of thing. One is nouns and one is a verb.

But still, for any real number that I “counted” I could match up a function, just like I did with

  • "5 that I counted" and
  • "function ƒ₅ = quintuple

So these two qualitatively different things are in bijection. (One can hope for insights by viewing things through one lens or the other, noun or verb version.)

This one-dimensional story can be upgraded to a multi-dimensional one where

  • lists of reals (3.1, √2, −2.1852, ..., 6) 

match to

  • many-to-one functions ƒ( list ) = 3.1•list[first] + √2•list[second] − 2.1852•list[third] + ... + 6•list[Nth].

Translating between the noun and verb viewpoints is then called musical isomorphism, represented with ♭ and ♯ symbols. Raising and lowering indices in a tensor is ♯ and ♭.