M03 · optional contextual support
Project a cloud and inspect what is lost
Connect unit directions, reconstruction, covariance, eigenvectors and low-rank intuition.
Start with arithmetic, percentages and basic algebra. Open any section directly. Practice and completion here do not change any AI course requirement or the original five-lesson mathematics pilot.
1. Try a concrete case
Project (2,1) onto the horizontal unit vector (1,0). The scalar coordinate is 2 and reconstruction is (2,0): the second coordinate has been lost.
2. Predict before changing a value
Rotate the projection direction and change vertical spread. Find the direction that retains most variation; explain why it could still erase a label-relevant small change. Name the quantity you expect to change and one you expect to stay fixed before using the controls.
3. Experiment
Change one quantity, follow every calculation
- Direction
- 1.00000, 0.00000
- Total centered energy
- 3.00000
- Retained fraction
- 0.66667
- Best axis
- horizontal
Vertical axis: Centered second measurement. Fixed scales; 0 points clipped for display. Lines join sampled values to aid reading; they do not create extra observations. Exact numerical values are below. Both coordinate axes use equal pixel scales so the displayed geometry matches the coordinate units.
| x | y | Coordinate on chosen axis | Reconstruction x | Reconstruction y |
|---|---|---|---|---|
| -2 | -1 | -2.00000 | -2.00000 | 0.00000 |
| -2 | 1 | -2.00000 | -2.00000 | 0.00000 |
| -1 | -1 | -1.00000 | -1.00000 | 0.00000 |
| -1 | 1 | -1.00000 | -1.00000 | 0.00000 |
| 0 | -1 | 0.00000 | 0.00000 | 0.00000 |
| 0 | 1 | 0.00000 | 0.00000 | 0.00000 |
| 1 | -1 | 1.00000 | 1.00000 | 0.00000 |
| 1 | 1 | 1.00000 | 1.00000 | 0.00000 |
| 2 | -1 | 2.00000 | 2.00000 | 0.00000 |
| 2 | 1 | 2.00000 | 2.00000 | 0.00000 |
For this symmetric cohort covariance is diag(2,b²). PCA picks its larger eigenvalue axis. A rank-one reconstruction keeps only the chosen component; discarded variation may matter to a later task. SVD of a diagonal matrix returns absolute diagonal scales, not a classification result.
Rotate the projection direction and change vertical spread. Find the direction that retains most variation; explain why it could still erase a label-relevant small change.
Controls save locally as practice. No timers, remote services or GPU are required. Maximum numerical work is a bounded small table or 256 bootstrap resamples.
4. Understand the calculation
Start with a calculation you can inspect
Project (2,1) onto the horizontal unit vector (1,0). The scalar coordinate is 2 and reconstruction is (2,0): the second coordinate has been lost.
Connect unit directions, reconstruction, covariance, eigenvectors and low-rank intuition. The experiment changes one named quantity at a time and shows the intermediate values, rather than asking you to accept a final score. Before changing a control, write down which value you expect to move and why.
Read the formula in words and symbols
coordinate = uᵀx; reconstruction = u(uᵀx), with uᵀu=1
x is a centered point and u is a unit direction. Transpose ᵀ changes a column into a row for the dot product. Retained energy divides projected squared length by original squared length.
A subscript names a coordinate, case or time index; it is not multiplication. A summation sign means add the terms for the stated indices. Parentheses identify which calculation happens first. Carry units through arithmetic: adding incompatible quantities or confusing a rate with a count changes the question.
Connect the visual and the numbers
Rotate the projection direction and change vertical spread. Find the direction that retains most variation; explain why it could still erase a label-relevant small change. The plot uses declared fixed axes, so a new setting does not silently rescale the apparent change. Any clipped values are counted and remain available in the numerical table. Do not infer an exact curve or a new observation from a line drawn between points.
Use it in an AI experiment
These ideas support C02, C04, C05. Use the return link when you entered from an AI lesson; its original section and experiment remain separate. A mathematical support calculation is a way to inspect an assumption, not evidence that a trained model generalizes.
Optional derivation and implementation depth
PCA chooses eigenvectors of centered covariance; C u=λu means direction u is scaled by λ. Equal eigenvalues make a preferred direction nonunique. SVD factors a matrix into orthogonal directions and nonnegative scales; truncating small scales gives a low-rank approximation, not a guarantee of downstream usefulness.
Recompute the worked example with a separate piece of arithmetic or a short program. State the domain and test a boundary case before using the formula elsewhere. Guided numerical practice, independent implementation and human review are different outcomes.
Optional implementation and research depth
PCA chooses eigenvectors of centered covariance; C u=λu means direction u is scaled by λ. Equal eigenvalues make a preferred direction nonunique. SVD factors a matrix into orthogonal directions and nonnegative scales; truncating small scales gives a low-rank approximation, not a guarantee of downstream usefulness.
Recompute one row independently, then test a boundary case and state which assumption it probes. Passing the supplied practice case does not establish independent implementation proficiency.
5. Apply it to a different case
Practice not yet passed; independent mastery and human review are not claimed.
6. Continue when useful
Completion not recorded. Skipping retains settings and practice; it never awards completion.
Used in C02, C04, C05. Browse every support topic and course connection.