Algebra II

Syllabus

• Vector spaces and subspaces; span and generating sets; linear combinations; linear dependence/independence; bases and coordinates.
• Exchange lemma and dimension: extending independent sets to bases; uniqueness of basis size; dimension of subspaces; sums and direct sums.
• Linear maps: kernels and images; injective/surjective criteria; rank–nullity theorem; matrix representation and change of basis.
• Rank of a matrix/linear map; row rank = column rank; Gauss elimination viewpoint; solvability of linear systems via rank conditions.
• Determinant as volume scaling; determinant of a linear map equals determinant of its matrix; invertibility ⇔ non-zero determinant.
• Eigenvalues/eigenvectors, eigenspaces, characteristic polynomial; diagonalizability criteria; minimal polynomial and Cayley–Hamilton (statements & consequences).
• Invariant subspaces; Jordan normal form (concept and uniqueness idea); powers of Jordan blocks (overview).
• Bilinear forms and quadratic forms; symmetric/Hermitian forms; Gram–Schmidt; Sylvester’s law of inertia; principal axes theorem (real case).
• Euclidean/unitary spaces: inner products, orthonormal bases, orthogonal complements; adjoint map; normal/unitary/orthogonal/symmetric operators (key properties).
• Tensor products (high-level overview); Kronecker product and relation to rank-1 decompositions (dyads).