Section 7.6 counts dimensions exactly. A table with columns and rank has a kernel of dimension , and the rank is the number of independent columns. A measured table is different. Chapter 6 showed that a computer stores most decimals with a small error, and a column that is a near-copy of another becomes an independent column once those errors are added. The exact rank of a large measured table is therefore often the full number of columns, even when the columns carry almost the same information. The chapter gives an exact count and no rule for deciding how many of those dimensions to trust. A fixed tolerance, a gap between successive pivot sizes, and a penalty charged for each column kept are three available habits, and they can disagree on the same table.
Take a position on this question: for a measured table, should a practitioner report the exact rank, or an effective dimension chosen by a stated rule?
In your answer: state a criterion for the effective dimension; apply it to a table you invent, with entries you can write down; say where the criterion stops working or what it quietly assumes; and name the evidence that would change your position.
Rubric. Tick each criterion your answer meets, then save the self-assessment. The platform does not grade it.
Worked solution
There is no single correct answer, so this is a model of the shape a complete answer takes, not a key. A level-3 answer chooses a table and makes the boundary visible. Take the three columns , and . Rounded to whole numbers, the first two columns both become , so the rounded table has rank 2 and a kernel of dimension 1. Kept exactly, the first two columns are not proportional, because times the first column is and the last entry is , so the exact rank is 3 and the kernel is only the zero vector. Neither count is wrong. The exact count answers a question about the written numbers; the effective count answers a question about what the measurements can distinguish.
A criterion has to say which question it answers. One possible criterion: rebuild each column from the others by elimination, and keep the column only if the leftover after the rebuild is larger than the measurement error. That criterion gives dimension 2 for the table above, because the leftover in the first two columns is below the error you declare. It assumes you know the error, and it changes its answer when you rescale a column, since a large unit makes every leftover large. A second criterion: keep only pivots above a fixed size. It is cheap and it is what many programs do, but the fixed size is a choice, not a property of the table. A third: charge a cost for each column and keep the columns whose contribution beats the cost. That turns the dimension into a decision about prediction rather than a property of the matrix, which may be the honest description.
The strongest objection is that a small difference can be the signal. Two measurements that differ by a thousandth may be nearly identical, or the thousandth may be the entire effect you are looking for. A criterion that discards the small direction also discards that effect. A strong answer takes a side on this: either report the exact rank and say that the effective dimension is a modelling choice made downstream, or report the effective dimension and say which error model and which cost justify it. Evidence that would change the position includes a task where the small direction predicts better than the large ones, or a noise model under which the small entries are demonstrably error. Without such evidence, the number of dimensions is a statement about the table and the rule, not about the world.