Interactive mathematics / information and recovery
What did your
summary forget?
Choose what you measure and what you want to know. Find an exact recovery rule, or two inputs your measurements cannot distinguish. Then repair the summary.
Exact rational arithmetic · 2–4 input coordinates · Runs in your browser
New here? Follow a four-step investigation
You need addition and the idea that a function gives one answer for each input. Start with this puzzle: (1, 0) and (0, 1) have the same total, but different first coordinates. What could a rule using only that total know?
Start with a question
1. Define the information
Read the rows as equations
A row gives the coefficient of each input coordinate. A zero skips that coordinate; a negative coefficient subtracts it. An unchecked measurement supplies no information to the recovery rule.
2. Test recovery
Loading the exact calculation…
- Independent measurements
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- Invisible directions
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- Extra measurements needed
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3. Make the summary sufficient
The rank calculation finds the smallest number of additional linear measurements needed for these questions.
Same information / different inputs
Move in an invisible direction.
An invisible direction w satisfies Rw = 0. Moving from x to x + tw leaves every enabled measurement unchanged.
Use the arrow keys on a slider, or the − and + buttons beside t. Home and End move to the ends. At t = 0 the inputs coincide. The pictures update without animation; exact values and descriptions are available below.
Which inputs share this summary?
Does the answer change as you move?
Solid line and circles: selected answer at x + tw. Dashed line: its value at the base input x. The vertical marker shows the current t. Graphs round fractions for display.
Read all nine graph samples as exact values
A flat graph along one direction alone does not prove recovery. The exact certificate tests every invisible direction. A nonzero slope supplies a collision.
Compare each input coordinate and measurement
Input coordinates change
Enabled measurements stay fixed
Filled marks: x. Outlined marks: x + tw. Plots approximate the exact fractions shown below.
| Quantity | At x | At x + tw |
|---|
Notation, everyday meanings, and access help
- Input x
- The full list of numbers you started with. A coordinate is one number in that list.
- Measurement y = Rx
- A weighted sum of input coordinates. Each enabled row of R supplies one number in the summary y.
- Question F
- A weighted sum you want to know. Each row of F is a separate question; F₁ and F₂ name different answers.
- Recovery rule G
- A rule that calculates the requested answers using only the summary. For linear recovery, F = GR says the measurement rows combine to give the question rows.
- Kernel / invisible direction w
- A change to the input that every enabled measurement erases: Rw = 0. A fiber is the whole set of inputs with one fixed summary.
- Rank and nullity
- Rank counts independent measurements. Nullity counts independent invisible directions. They add up to the number of input coordinates.
- Certificate
- An exact identity you can check: either weights that reconstruct the question, or two indistinguishable inputs with different answers.
Tab moves between controls. Space toggles a measurement; Enter activates a button. Arrow keys adjust sliders. Select a question button to inspect its certificate. Guided-step buttons replace the experiment and keep focus; “Next guided step” moves focus to the new explanation.
Every graph has a visible description and a table of exact values. The two-coordinate picture shows a full fiber; higher-dimensional experiments use the answer-versus-t graph. Line styles and mark shapes distinguish series as well as color. Wide tables can be focused and scrolled with arrow keys.
Results update as you type. Invalid entries are marked and described; the last valid calculation stays visible until you fix them. The written worked example is available without JavaScript. See accessibility and reading options for the wider collection.
Inspect the kernel and the row reduction
See the row operations
The same row operations give a matrix E with E R_active = R_reduced. The recovery coefficients are formed from E, so their product with the original enabled rows can be checked directly.
Go deeper: why do these certificates settle the question?
Let R: ℝⁿ → ℝᵐ be the enabled measurements, and let F: ℝⁿ → ℝᵏ be the requested linear answers. Exact recovery from R for every input is possible if and only if ker R ⊆ ker F. Equivalently, every row of F lies in the row space of R, so a matrix G satisfies F = GR.
If F = GR, the rule y ↦ Gy recovers F(x) from y = Rx. If a vector w has Rw = 0 but Fw ≠ 0, then x and x + w have identical measurements and different answers. This rules out every recovery function of those measurements, including nonlinear functions.
To see the converse directly, take a basis of ker R and extend it to a basis of ℝⁿ. When F vanishes on ker R, assign G(Rv) = Fv on the images of the added basis vectors. These images form a basis of im R. Extend G linearly to ℝᵐ. Then GR = F.
The minimum number of extra scalar linear measurements is rank([R; F]) − rank(R), where [R; F] means stacking the rows. Each added row increases rank by at most one. Adding a basis of the missing question directions attains this bound. “Build a sufficient summary” keeps an independent subset of the enabled rows and adds the required question rows; redundant and disabled rows are removed.
These statements concern unrestricted real inputs and exact linear measurements. The displayed matrices have rational entries and are reduced using exact integer fractions. The laboratory does not model noisy measurements, restricted input sets, or approximate recovery. Its calculations add no Lean coverage to the manuscripts.