The Shadow and the Thing
Last timeOne Map After Another
Every map reaches a limited set of outputs and destroys a definite set of inputs, and the two sizes add up to the input dimension, which is the whole bookkeeping of rank.
A map can only produce outputs that are mixtures of its columns, because that is
what multiplication does. If a matrix has three columns, every output it will ever
produce is three numbers worth of mixing applied to those three particular
destinations. Nothing else is reachable, no matter what input you feed in.
That observation sounds modest and is not. It means the set of possible outputs is
fixed by the matrix before any input arrives, and it can be much smaller than the
output space looks. A map from the plane to the plane whose two columns point in
the same direction can only ever produce outputs along that one direction. The
output space is a plane; the reachable set is a line inside it. Half the apparent
freedom was never there.
What can be reached
Call that reachable set the image of the map. It is always flat and it always
contains the origin, because feeding in the zero vector gives zero out. Its size
is the number of genuinely different directions among the columns, which is
usually smaller than the number of columns and never larger.
What is destroyed
Now the other side. Ask which inputs the map sends to zero. There is always at
least one, the zero vector itself, and that case is uninteresting. The question is
whether a non-zero input is sent to zero.
If one is, the map has lost information, and the argument is short. Suppose some
non-zero vector is sent to zero. Take any input at all and add that vector to it.
By additivity the output is unchanged, because the added piece contributed
nothing. So two different inputs produce the same output, and from the output
there is no way to tell which one you started with. The map is not reversible, and
no cleverness in the arithmetic will make it so.
Running the argument the other way is just as clean. If two different inputs land
on the same output, their difference lands on zero. So confusion between inputs
and non-zero vectors sent to zero are the same phenomenon viewed twice.
The lesson stops here
6 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