Undo, If You Can
Last timeWhat a Map Throws Away
An inverse is the map that undoes another one, it exists only when nothing was destroyed and everything is reachable, and solving a system is never the same job as building one.
The last lesson named the two things attached to any map: what it can reach, and
what it destroys. This lesson spends both of them, because the question of whether
a map can be run backwards has exactly those two answers inside it.
An inverse is another map. Nothing more exotic. Given a map that takes inputs to
outputs, an inverse takes outputs back to inputs in such a way that doing one and
then the other leaves every vector exactly where it started. Since doing one map
after another is a product, the condition is a product equation.
- the original map
- the map that undoes it, if such a map exists
- any input, returned unchanged after the round trip
- any output, also returned unchanged after the round trip the other way
When it exists
Put the two failures side by side. The first is destruction. If some non-zero
input goes to zero, then two different inputs share an output, and an undoing map
asked about that output would have to return both. There is no such map. The
second is unreachability. If some vector in the output space is never produced,
then an undoing map asked about that vector has nothing to return.
Rule out both and the inverse exists, and it is unique. For a square matrix the
counting law collapses the two conditions into one. If nothing is destroyed then
the destroyed count is zero, so the rank equals the number of columns, which for a
square matrix equals the number of rows, which means the reachable set fills the
output space. Nothing is lost and everything is reached, from a single assumption.
This is why full rank is the condition people quote. It is not an extra
requirement on top of being square; it is the whole requirement, and squareness is
what lets one half of it imply the other.
Solving is a different job
On paper, solving a system is written as multiplying the target by the inverse,
and that notation is perfectly correct. It is also a poor instruction for actually
getting a number.
Building the inverse means answering the question for every possible target at
once. Solving the system means answering it for the one target you have. The
second is a much smaller question, and the arithmetic reflects that: for a map of
any size the direct solve costs roughly a third of what forming the inverse costs,
and then multiplying by the formed inverse costs more again.
The lesson stops here
4 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents