Each Column Is a Destination
Last timeSomething That Moves Vectors
The columns of a matrix are the places the axes land, which turns reading a matrix into looking at a short list of destinations and writing one into a design decision.
The last lesson ended with a formula showing that a linear map is completely
determined by where it sends the axes. This lesson collects the consequence,
which is the single most useful habit in the subject.
Take the simplest possible input: a one in the first slot and zeros everywhere
else. Running it through the combination formula, every term with a zero
coefficient vanishes and one term survives. The output is wherever the first axis
went. Do the same with a one in the second slot and you get the second
destination.
Those destinations are the columns. That is what the grid is: a list of
destinations, written vertically and set side by side.
- the output, a combination of the columns with the input coordinates as weights
- the first column, which is where the first axis lands
- a coordinate of the input, acting as the amount of that column to use
- the last column, one per axis and no more
Writing a map down on purpose
The column reading turns matrix construction from memorisation into a short
decision. Decide where each axis should go, write those down as columns, and the
matrix is finished.
| first column, x part | first column, y part | second column, x part | second column, y part | |
|---|---|---|---|---|
| stretch x by two, y by t | 2 | 0 | 0 | 3 |
| quarter turn anticlockwi | 0 | 1 | -1 | 0 |
| shear, leaning x to the | 1 | 0 | 1 | 1 |
| flatten everything onto | 1 | 0 | 0 | 0 |
| step | step | decision | result so far |
|---|---|---|---|
| 1 | name the map | reflect the plane in the line at forty f | nothing written yet |
| 2 | send the first axis | the x axis reflects onto the y axis | first column is zero then one |
| 3 | send the second axis | the y axis reflects onto the x axis | second column is one then zero |
| 4 | write it down | set the two destinations side by side | the matrix that swaps the two coordinate |
| 5 | check it | apply it twice and see that everything r | reflecting twice is doing nothing, as it |
The same method scales to any number of dimensions and to maps that have no
name. A map that keeps the first three coordinates of a long embedding and
discards the rest has columns that are the first three axes of the output space
followed by a great many zero columns, and writing that down takes no thought at
all once the rule is in hand. A map that averages two features into one has a
column of one half in both of the relevant places. In each case the description
in words translates directly into a list of destinations, with no formula to
recall and nothing to get backwards.
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