Posts tagged with tangent

## Vector Duals and Musical Isomorphism in 1-D

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`.

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,∞).

To jump between the two worlds you use the reciprocal map flip(■)≝1/■. T

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 ♭.