Linear Algebra II
A proof-based second course in linear algebra: abstract vector spaces and subspaces, bases and dimension, linear maps with their kernels and images and the rank–nullity theorem, matrices of linear maps, change of basis and similarity, determinants and eigenvalues of operators, diagonalization with applications to recurrences, Markov chains and differential equations, invariant subspaces, the Cayley–Hamilton theorem and minimal polynomials, the Jordan canonical form, inner product spaces, Gram–Schmidt and orthogonal complements, projections, least squares and the QR factorization, the spectral theorem and quadratic forms, and a capstone on the singular value decomposition, low-rank approximation and principal component analysis, with runnable Python examples throughout.
MATH-221 Year 2 3 credits 12 lessons · 4 h 38 min
Offered by HSM University · Mathematics & Statistics
What you'll learn
- State the vector space axioms, verify them for spaces of tuples, polynomials, matrices, functions and sequences, and use the subspace test to identify subspaces, intersections, sums and direct sums.
- Prove and apply the replacement theorem, the invariance of dimension and the dimension formula for sums, and compute bases, dimensions and coordinate vectors with exact arithmetic.
- Determine the kernel and image of a linear map, prove and apply the rank–nullity theorem, and recognize injective, surjective and invertible maps, isomorphisms and quotient spaces.
- Build the matrix of a linear map relative to chosen bases, convert coordinates between bases, relate composition to matrix multiplication, and use similarity invariants to compare matrices.
- Characterize the determinant as the unique alternating multilinear normalized function, define eigenvalues and eigenspaces for operators on any vector space, and distinguish algebraic from geometric multiplicity.
- Decide whether an operator is diagonalizable, compute powers and exponentials of diagonalizable matrices, and apply them to linear recurrences, Markov chains and systems of differential equations.
- Work with invariant subspaces and triangular forms, prove and apply the Cayley–Hamilton theorem, and compute the minimal polynomial and use it to test diagonalizability.
- Find the Jordan canonical form of a matrix from ranks of powers, compute powers and exponentials of Jordan blocks, and explain why the Jordan form is unsuitable for floating point computation.
- Verify inner product axioms, prove the Cauchy–Schwarz and triangle inequalities, construct orthonormal bases by Gram–Schmidt, and work with orthogonal complements and adjoints.
- Characterize orthogonal projections, apply the best approximation theorem in function spaces, and solve least squares problems with the normal equations and, preferably, the QR factorization.
- Prove the spectral theorem for symmetric matrices, state its complex version for normal operators, find principal axes, classify quadratic forms and use the Rayleigh quotient.
- Derive the singular value decomposition, read the four fundamental subspaces from it, and apply it to low-rank approximation, the pseudoinverse, condition numbers and principal component analysis.
Course content
12 lessons · 4 h 38 min total · final exam
- 1 Vector spaces and subspaces 25 min · quiz
- 2 Span, independence, bases and dimension 24 min · quiz
- 3 Linear transformations, kernel and image 24 min · quiz
- 4 Matrices of linear maps and change of basis 22 min · quiz
- 5 Determinants and eigenvalues revisited 23 min · quiz
- 6 Diagonalization and applications 21 min · quiz
- 7 Invariant subspaces, Cayley–Hamilton and the minimal polynomial 24 min · quiz
- 8 Jordan canonical form 23 min · quiz
- 9 Inner product spaces and orthogonality 23 min · quiz
- 10 Orthogonal projections, least squares and QR 23 min · quiz
- 11 The spectral theorem and quadratic forms 22 min · quiz
- 12 Capstone: the singular value decomposition and its applications 24 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
A proof-based second course in linear algebra: abstract vector spaces and subspaces, bases and dimension, linear maps with their kernels and images and the rank–nullity theorem, matrices of linear maps, change of basis and similarity, determinants and eigenvalues of operators, diagonalization with applications to recurrences, Markov chains and differential equations, invariant subspaces, the Cayley–Hamilton theorem and minimal polynomials, the Jordan canonical form, inner product spaces, Gram–Schmidt and orthogonal complements, projections, least squares and the QR factorization, the spectral theorem and quadratic forms, and a capstone on the singular value decomposition, low-rank approximation and principal component analysis, with runnable Python examples throughout.
Part of the Science stream, typically taken in year 2.