Answers · The mathematics underneath

Does a subscript like a_ij mean a times i times j?

In shortNo, a subscript marks which row and column an entry belongs to rather than multiplying by the letters underneath.

No. In \(a_{ij}\) the letters \(i\) and \(j\) are not factors. They say which row and which column of the table \(a\) is being named.

a_{ij} = \text{the entry in row } i \text{ and column } j \text{ of } a.

FIG 1

column 1	column 2	column 3
5	2	9
1	7	3
8	4	6
2	2	1

Reading the positions

Take the entry \(a_{23}\). The first subscript is 2, so row 2. The second is 3, so column 3. In the table that entry is 3. Now take \(a_{32}\): row 3, column 2, which is 4. If the subscripts were multiplication, \(a_{23}\) would be \(a\cdot2\cdot3\) and \(a_{32}\) would be \(a\cdot3\cdot2\). Both products equal \(6a\), because \(2\cdot3=3\cdot2=6\). The two entries would have to be equal. They are 3 and 4, so they are not. The ordering of the subscripts is doing real work. The bare name \(a\), with no subscript, means the whole table at once, which is why a line that uses both \(a\) and \(a_{ij}\) is drawing a distinction between the collection and one of its entries.

The subscripts also tell you which direction a sum runs.

\begin{aligned} \sum_{i=1}^{4} a_{i2} &= a_{12}+a_{22}+a_{32}+a_{42} = 2+7+4+2 = 15,\\ \sum_{j=1}^{3} a_{2j} &= a_{21}+a_{22}+a_{23} = 1+7+3 = 11. \end{aligned}

In the first sum the row index \(i\) is bound by the sum sign. The rows are added, and the column index \(j=2\) is free, so the result is one number for column 2. In the second sum the column index \(j\) is bound and the row index \(i=2\) is free, so the result is one number for row 2. Changing which letter sits under the sum sign changes a column total into a row total, and the rest of the line can look identical.

The guess that \(a_{ij}\) means \(a\times i\times j\) fails for two separate reasons. First, \(a\) is the whole table, and multiplying a table by two numbers is not what the notation is doing. Second, even if \(a\) were a plain number, multiplication is commutative: \(a\cdot2\cdot3\) and \(a\cdot3\cdot2\) are the same number. Entries of a table are not required to be the same, and \(a_{23}=3\) while \(a_{32}=4\). A notation that forced those to be equal could not describe an ordinary table. The subscript is a label for a position, not a factor in a product.

When a subscript is a name

Not every subscript counts. \(x_{\text{train}}\) and \(x_{\text{test}}\) name which collection is meant; \(\theta_t\) names a round; \(\pi_\star\) marks a special one. The test is whether adding one would mean anything. In \(x_i\), \(x_{i+1}\) is the next item. In \(x_{\text{train}}\), \(x_{\text{train}+1}\) is not a next item. Labels like these cannot be summed over, because there is no ordered list to run along. The same mark can look identical in both cases, so the classification comes from what the symbol was declared to be, not from the shape of the subscript.

x_{\text{train}}, \qquad x_{\text{test}}, \qquad \theta_t, \qquad \pi_\star

Where the marks above the line differ

The mark above the line is a separate ambiguity. \(x^2\) is a power; \(x^{(2)}\) is often the second item or the second layer; \(A^\top\) swaps rows and columns of a table. The brackets are a hint, not a rule. What settles it is the kind of object underneath. A power needs something that can be multiplied by itself, a transpose needs a table, and an index needs a collection to pick from. A subscript is more consistent than this: one subscript picks an item out of a collection, two pick an entry out of a table in row then column order, and a bare name without a subscript means the whole collection. The order is a convention, but once fixed it is content: it is why \(a_{23}\) and \(a_{32}\) are allowed to be different.

Worth keeping

A subscript is a label for position, not a factor in a product.
In a two-subscript symbol the first subscript is the row and the second is the column.
Subscripts that are words or stars are labels and cannot be summed over.

Where this goes next

This question is one step of Reading the Notation, a course that works through its subject from the beginning. The page that teaches this part of it is Numbering Things, and it reads the whole way through whether or not you have an account.

What an account changes is the practice: problems standing under every idea, marked with a reason rather than a tick, in a copy of the course that is yours.

Read that page The whole course