Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook defines least-squares solutions of Ax=b by projecting b onto the image of A. It derives the normal equations AᵀAx=Aᵀb and uses them to solve inconsistent systems.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook defines inner products on real vector spaces, Euclidean spaces, and orthogonality. Examples include trace-based products on matrices and integral products on function spaces, followed by classification exercises.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook proves that the orthogonal projection of a vector onto a subspace is its best least-squares approximation in that subspace. Students construct orthogonal bases and compute closest vectors with an interactive Gram-Schmidt tool.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook constructs the QR factorization of a matrix with independent columns using Gram-Schmidt. It then applies Q and the upper-triangular factor R to compute least-squares solutions efficiently by backward substitution.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook derives norms and distances from inner products and presents the Cauchy-Schwarz and triangle inequalities. It also defines angles and develops generalized Pythagorean identities through visual examples and exercises.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook defines projection onto a vector and develops the Gram-Schmidt process for turning an independent family into an orthogonal or orthonormal one with the same span. Interactive exercises guide each subtraction and normalization step.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook defines the characteristic polynomial det(A−tI), explains its invariance under similarity, and connects its roots with eigenvalues. Exercises compute characteristic polynomials and spectra of matrices.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook defines the eigenspace associated with an eigenvalue and identifies it with the kernel of A−λI. Students compute eigenspace bases and use visualizations to observe invariant lines and planes.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook extends eigenvalue analysis and diagonalization from real to complex vector spaces. It uses the fundamental theorem of algebra, conjugate roots, and complex eigenspaces to diagonalize real matrices with nonreal eigenvalues.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook characterizes diagonalizable transformations by factorization of the characteristic polynomial and equality of algebraic and geometric multiplicities. It gives a practical method for constructing P and D from eigenvector bases.