ISEGORIA / MATH ENCYCLOPEDIA
Linear algebra: the geometry of transformations
Coordinates, area, eigenvectors, projection, and singular values.
Before you begin: Vectors and elementary algebra
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Basis and coordinates
Drag the two circular basis handles or use the sliders. The highlighted physical vector stays at (1,1), while its coordinates in the new basis change. The columns of a matrix are the images of the standard basis.
Worked example. For B=diag(2,1), the coordinates of (1,1) are (1/2,1).
Watch out. Dependent columns do not form a basis; an inverse does not exist.
What happens when the two columns coincide?
The plane collapses to a line. Coordinates are no longer unique when a solution exists.
2. Determinant and rank
The unit square becomes a parallelogram. Its area is the absolute determinant; the sign records orientation. Set one scale to zero to see rank fall.
Worked example. For diag(-2,1), area doubles and orientation reverses.
Watch out. A determinant of one preserves area, but need not preserve lengths or angles.
Can a shear change shape without changing area?
Yes. The matrix with rows (1,s) and (0,1) has determinant one for every s.
3. Eigenvectors and iteration
Rotate the eigenbasis and change its two eigenvalues. Repeated application scales each eigenvector component separately. Dashed lines mark the invariant directions.
Worked example. With eigenvalues 1.2 and 0.5, one component grows while the other decays.
Watch out. This laboratory uses real symmetric matrices. A real rotation need not have a real eigenvector.
If both eigenvalues have magnitude below one, where does every orbit go?
To the origin, because both scalar powers tend to zero.
4. Least squares as projection
Rotate a line through the origin. The closest point on it to v=(2,1) is the orthogonal projection. The residual is perpendicular to the line.
Worked example. Projecting (2,1) onto the x-axis gives (2,0), with residual (0,1).
Watch out. A fitted projection minimizes squared error, not necessarily every other loss function.
When is the residual zero?
When the vector lies on the chosen line.
5. Singular values and rank
Rotate the input, stretch along two perpendicular axes, then rotate the output. The image of the unit circle is an ellipse. Removing the smaller stretch gives a rank-one approximation.
Worked example. For diag(2,0.5), the best rank-one approximation is diag(2,0), with operator-norm error 0.5.
Watch out. Singular values are nonnegative; they are not generally the eigenvalues of A.
What happens when the second singular value is zero?
The ellipse collapses to a segment and the rank is one.