Unit 3 of 10
Determinants and Bilinear Forms
- 3 lessons
- Videos 13 to 15
This unit builds the determinant from a few defining properties and shows which numbers attached to a matrix stay the same when the basis changes.
The determinant is the unique function of the columns that is multilinear, alternating and equal to one at the identity matrix, and its product rule follows from that uniqueness. Trace and determinant are unchanged by similarity, while the matrix of a bilinear form changes by congruence, so the same square array can follow two different coordinate laws.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Determinants from Alternating Forms
Determinants and Multilinear Forms
Why a multilinear, alternating form that equals one at the identity must be the determinant, how the sign of a permutation enters its formula, and how uniqueness proves the product rule.
Video 13
Coordinate Invariants and Bilinear Forms
Trace, Determinant and Similarity
Why similar matrices share their trace, determinant and characteristic polynomial, proved from the cyclic property of the trace and the product rule, and the real volume factor of a complex matrix.
Video 14
Bilinear Forms and Congruence
How a bilinear form is stored in a matrix, why a change of basis acts on that matrix by congruence rather than similarity, and how polarization recovers a symmetric form from its quadratic form.
Video 15