Linear Transformations and Matrices
Stub. Linear maps, null spaces/ranges, matrix representation and change of basis, composition as matrix multiplication, invertibility and isomorphisms, dual spaces, products/quotients of vector spaces, and the computational machinery of elementary row operations, rank, and solving systems of linear equations.
The functions that respect vector-space structure, and the finite arrays that represent them once bases are fixed. Covers the structural theory — null space/range, the fundamental theorem of linear maps, invertibility and isomorphism, change-of-basis matrices, dual spaces and dual maps, products and quotients of vector spaces — alongside the computational counterpart: elementary matrices and row reduction, rank, and solving systems of linear equations both theoretically (solution-set structure) and computationally (Gaussian elimination). Friedberg–Insel–Spence’s optional section on constant-coefficient linear ODEs is included as a worked application of this machinery.
Outline (TODO — flesh out each)
- Linear transformations, null spaces, and ranges
- The matrix representation of a linear transformation
- Composition of linear transformations and matrix multiplication
- Invertibility and isomorphisms
- The change of coordinate matrix
- Products and quotients of vector spaces
- Dual spaces; the dual map and its matrix
- Application: homogeneous linear differential equations with constant coefficients
- Elementary matrix operations and elementary matrices
- The rank of a matrix and matrix inverses
- Systems of linear equations — theoretical aspects
- Systems of linear equations — computational aspects (row reduction)