Courses / Mathematics / Linear Algebra II: operators and spectral structure Mathematics · unit version 1 Linear Algebra II: operators and spectral structure A finite-dimensional operator sequence over R and C, with quotient and dual constructions, canonical forms, spectral proofs and singular-value analysis. Written proofs require review; publication alone does not establish semester equivalence.
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Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
If a linear map from R⁵ has rank 3, what is its nullity? Abstract operator arguments reuse dimension, kernel and image rather than only row reduction.
Compare reasoning and find a refresher Rank–nullity gives 5=3+nullity, so the kernel has dimension 2. The domain dimension is the one used in this identity.
Review rank–nullity
Does a repeated eigenvalue guarantee enough eigenvectors to diagonalize a matrix? Generalized eigenvectors are needed when ordinary eigenspaces are too small.
Compare reasoning and find a refresher No. The matrix with rows (1,1) and (0,1) has eigenvalue 1 twice but its eigenspace is only the span of (1,0), so it is not diagonalizable.
Review diagonalization
What makes the residual of an orthogonal projection special? Orthogonality connects adjoints, least squares and singular values.
Compare reasoning and find a refresher The residual is perpendicular to the entire target subspace. This yields the Pythagorean identity and the least-distance property, not merely a small residual in one example.
Review orthogonal projections
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Operators: structure, geometry and justified decompositions
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Lesson sequence Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Lesson details and objectives Work with polynomial and matrix spaces, then distinguish a map from its coordinate matrix.
Construct bases in finite-dimensional function and matrix spaces. Prove linearity from the two defining operations. Transport a map through chosen domain and codomain bases. 6 practice tasks.
Lesson details and objectives Build complements, prove rank-nullity, and distinguish an internal direct sum from an arbitrary sum.
Construct a complement by extending a basis. Prove rank-nullity through a transported basis. Test uniqueness in an internal direct sum. 6 practice tasks.
Lesson details and objectives Construct quotient spaces and prove the first isomorphism theorem for linear maps.
Check quotient operations are independent of representatives. Construct a quotient basis and compute its dimension. Prove the induced map V/ker T→im T is an isomorphism. 8 practice tasks.
Lesson details and objectives Build the dual basis, annihilators and the pullback of a linear map.
Construct coordinate functionals and a dual basis. Prove the annihilator dimension formula. Relate the dual map to transpose matrices and kernels. 8 practice tasks.
Lesson details and objectives Connect invariant subspaces, quotient maps and block-triangular matrices.
Test invariance by checking basis images. Derive block-triangular matrices from invariant subspaces. Distinguish invariant complements from arbitrary complements. 8 practice tasks.
Lesson details and objectives Relate characteristic polynomials, multiplicities and triangular forms to the scalar field.
Prove similarity preserves the characteristic polynomial. Distinguish algebraic and geometric multiplicities. Explain the split-field hypothesis for triangularization. 6 practice tasks.
Lesson details and objectives Construct annihilating polynomials and connect square-free splitting to diagonalizability.
Prove existence and divisibility of the minimal polynomial. Compute minimal polynomials from simple blocks. Prove the distinct-root criterion for diagonalizability. 6 practice tasks.
Lesson details and objectives Prove Cayley–Hamilton and reduce powers and inverses modulo a polynomial relation.
Give a valid coefficient proof of Cayley–Hamilton. Reduce high powers using a characteristic relation. Construct inverses through polynomial identities when permitted. 8 practice tasks.
Lesson details and objectives Use Bézout identities to decompose a space and define generalized eigenspaces under splitting.
Prove a coprime-kernel decomposition using Bézout. Compute generalized eigenspaces by powers of T−λI. State how the minimal polynomial controls each primary component. 6 practice tasks.
Lesson details and objectives Construct nilpotent chains, recover block sizes from kernel growth, and state Jordan uniqueness precisely.
Translate a generalized-eigenvector chain into a Jordan block. Recover block counts from successive nullities. Justify chain-basis existence and identify the required field hypothesis. 6 practice tasks.
Lesson details and objectives Develop real and complex inner products, norms and Cauchy–Schwarz with a consistent convention.
Compute an inner product linear in its first argument. Prove Cauchy–Schwarz using a nonnegative squared norm. Distinguish bilinear products from positive Hermitian forms. 6 practice tasks.
Lesson details and objectives Prove orthogonal decomposition, best approximation and finite-dimensional Riesz representation.
Construct an orthogonal projection in an orthonormal basis. Prove uniqueness and best approximation by a Pythagorean identity. Explain conjugate-linearity in the Riesz correspondence. 8 practice tasks.
Lesson details and objectives Construct adjoints and relate their kernels and ranges to orthogonal complements.
Derive the conjugate-transpose formula in orthonormal bases. Prove kernel/range orthogonality identities. Distinguish an adjoint from an algebraic transpose in general coordinates. 8 practice tasks.
Lesson details and objectives Prove complex Schur triangularization and the orthonormal spectral theorem for normal operators.
Prove Schur triangularization by orthonormal induction. Show an upper-triangular normal matrix is diagonal. State the real/complex scope of the normal spectral theorem. 6 practice tasks.
Lesson details and objectives Prove real spectra and orthogonal diagonalization, then derive Rayleigh bounds and extremizers.
Prove eigenvalues of a self-adjoint operator are real. Derive Rayleigh quotient bounds and equality conditions. Relate real symmetric operators to orthogonal diagonalization. 8 practice tasks.
Lesson details and objectives Classify Hermitian forms, construct positive square roots, and distinguish congruence from similarity.
Characterize positivity through eigenvalues. Construct and prove uniqueness of the positive square root. Explain inertia and the role of congruence in quadratic forms. 8 practice tasks.
Lesson details and objectives Characterize unitary maps, prove simultaneous spectral decompositions and derive polar decomposition.
Prove equivalent isometry and unitary conditions. Prove simultaneous diagonalization of commuting normal complex operators. Construct polar decomposition and explain nonuniqueness on kernels. 8 practice tasks.
Lesson details and objectives Derive SVD from the spectral theorem for A*A and interpret singular values geometrically.
Construct singular vectors from eigenvectors of A*A. Explain the image of a unit sphere including rank-deficient cases. Distinguish singular values from eigenvalues and prove the operator norm formula. 6 practice tasks.
Lesson details and objectives Construct the Moore–Penrose inverse, prove minimum-norm least squares and derive optimal truncated SVD.
Construct a pseudoinverse from positive singular values. Characterize all least-squares solutions and the unique minimum-norm one. Prove the operator-norm error bound for truncated SVD. 6 practice tasks.
Lesson details and objectives Derive spectral condition numbers, perturbation bounds and the costs of normal equations.
Derive the 2-norm condition number from singular values. Prove a right-hand-side perturbation bound and interpret its worst-case nature. Explain why normal equations square conditioning and choose a suitable decomposition. 6 practice tasks.
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