Why the Multiplication Rule Looks Like That

Last timeReading a Matrix Off Its Columns

Matrix multiplication is doing one map and then another, which forces the strange entry rule, explains why order matters, and makes the cost of a chain a real decision.

Of everything in elementary linear algebra, the rule for multiplying two matrices

is the part that looks most invented. Walk along a row, down a column, multiply

and add, repeat. Nothing about the grid suggests it, and nobody who learns it that

way can say why.

From the map view it is not a rule at all. It is a consequence.

FIG 1Two maps, two orders

This is the whole case against commuting, in a picture. The two routes apply the same pair of maps and arrive somewhere different, so the composite maps are different, so the two products are different matrices. No algebra is required to see it.

Why the rule is what it is

Start from the column reading. The composite map is applied by running the first

map and then the second, so the composite sends the first axis to wherever the

second map sends the first map's first column.

Now unpack that. The first map's first column is a vector of coordinates. The

second map, handed those coordinates, mixes its own columns using them as

amounts. Writing out that mixture produces a sum over an index, with one factor

from each matrix, and that sum is exactly the famous rule.

FIG 2Composition, stated plainly

: apply the right hand map first, which is why products are read right to left
: apply the left hand map to whatever came out
: the single matrix that does both in that order, which is what the product means

This is the definition, and the entry formula is derived from it rather than the other way round. It also explains the notation that trips everyone up: in a product the rightmost matrix acts first, because it is the one nearest the vector. The order on the page runs opposite to the order in time.

FIG 3The entry rule, now unsurprising

: the ith component of where the composite sends the kth axis
: the sum over the intermediate space, which is where the shared index comes from
: the ith component of where the second map sends the jth axis
: the jth amount in the kth destination of the first map

The index that is summed over is the one living in the intermediate space, which is why the inner dimensions must agree: there has to be an intermediate space for the two maps to meet in. Every oddity of the rule is a feature of composition rather than a convention of notation.

It is worth pausing on why a composite of two linear maps is linear at all,

since the whole construction depends on it. Hand the composite a sum. The first

map splits it, by additivity. The second map splits the two pieces again, for the

same reason. Scaling passes through both maps in the same way. So the composite

obeys both rules, which means it has a matrix of its own, which is the object the

product names. Without that short argument there would be nothing for the product

to be.

The lesson stops here

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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 plan The contents

This 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

Why the Multiplication Rule Looks Like That

Why the rule is what it is

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