Program
History and progress transfer
Session summary
Homework — optional, outside the timer
Theory of all steps of the program by blocks. Open a topic to read it.
How a session goes
A session is 10–90 minutes: a warm-up on what you've learned, new steps with short theory and problems, practice. Each problem gives two tries: 100 points on the first, 60 on the second; after the second mistake the correct answer is shown. You move on in the program if 70 % of the problems are solved.
Problems with numbers are generated anew every time a step is repeated. A fractional answer can be entered with a comma, a point or as a fraction: 0,5, 0.5, 1/2. In Practice any topic is available without a timer.
About the course
The n×n determinant and its properties, rank and Kronecker–Capelli, fundamental systems of solutions, vector spaces and bases, linear operators, kernel and image, eigenvalues and diagonalization, Gram–Schmidt, orthogonal matrices, quadratic forms and Sylvester's criterion, SVD. Builds on the “Matrices” mini-course.
Course program: 32 lessons
n×n determinants
- Expansion along a row
- Properties of the determinant
- Determinants: more
- Checkpoint: determinants
Rank and systems of equations
- Rank of a matrix
- Rank: more
- The Kronecker–Capelli theorem
- Homogeneous systems and the fundamental system of solutions
- Checkpoint: rank and systems
Vector spaces
- Vector space
- Linear dependence
- Basis and dimension
- Change of basis
- Checkpoint: spaces and bases
Linear operators
- A linear operator and its matrix
- Kernel and image
- The operator's matrix in another basis
- Operators: more
- Checkpoint: linear operators
Eigenvalues
- Eigenvalues
- Eigenvectors
- Eigenvalues and eigenvectors: more
- Diagonalization
- Checkpoint: eigenvalues
Euclidean spaces
- Dot product
- Gram–Schmidt orthogonalization
- Orthogonal matrices
- Checkpoint: Euclidean spaces
Quadratic forms and SVD
- Quadratic forms
- Sylvester's criterion
- Singular value decomposition (overview)
- Checkpoint: forms and SVD