The rank-nullity theorem in linear algebra says that dimensions either get
- thrown in the trash
- or show up
after the mapping.
By “the trash” I mean the origin—that black hole of linear algebra, the
/dev/null, the ultimate crisscross paper shredder, the ashpile, the wormhole to void and cancelled oblivion; that country from whose bourn no traveller ever returns.
The way I think about rank-nullity is this. I start out with all my dimensions lined up—separated, independent, not touching each other, not mixing with each other.
||||||||||||| like columns in an Excel table. I can think of the dimensions as separable, countable entities like this whenever it’s possible to rejigger the basis to make the dimensions linearly independent.
I prefer to always think about the linear stuff in its preferably jiggered state and treat how to do that as a separate issue.
So you’ve got your
172 row × 81 column matrix mapping 172→ separate dimensions into →81 dimensions. I’ll also forget about the fact that some of the resultant →81 dimensions might end up as linear combinations of the input dimensions. Just pretend that each input dimension is getting its own linear λ stretch. Now linear just means multiplication.
Linear stretches λ affect the entire dimension the same. They turn a list like
[1 2 3 4 5] into
[3 6 9 12 15] (λ=3). It couldn’t be into
[10 20 30 − 42856712 50] (λ=10 except not everywhere the same stretch=multiplication).
Also remember – everything has to stay centred on
0. (That’s why you always know there will be a zero subspace.) This is linear, not affine. Things stay in place and basically just stretch (or rotate).
So if my entire 18th input dimension
[… −2 −1 0 1 2 3 4 5 …] has to get transformed the same, to
[… −2λ −λ 0 λ 2λ 3λ 4λ 5λ …], then linearity has simplified this large thing full of possibility and data, into something so simple I can basically treat it as a stick
If that’s the case—if I can’t put dimensions together but just have to λ stretch them or nothing, and if what happens to an element of the dimension happens to everybody in that dimension exactly equal—then of course I can’t stick all the 172→ input dimensions into the →81 dimension output space.
172−81 of them have to go in the trash. (effectively, λ=0 on those inputs)
So then the rank-nullity theorem, at least in the linear context, has turned the huge concept of dimension (try to picture 11-D space again would you mind?) into something as simple as counting to 11