Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook defines orthogonal linear transformations and matrices through norm preservation. It proves injectivity and angle preservation, then uses interactive two-dimensional transformations to distinguish orthogonal maps from deformations.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook defines quadratic forms on Rⁿ using symmetric matrices and connects level sets to conics and quadrics. Exercises translate polynomial expressions into the symmetric matrix representation xᵀAx.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook connects symmetric matrices with orthogonal diagonalization. It proves that eigenvectors associated with distinct eigenvalues are orthogonal and presents the spectral theorem for real symmetric matrices.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook applies orthogonal diagonalization to quadratic forms through the principal axis theorem. Students find the orthogonal change of variables that removes cross terms and identify the principal axes from eigenvectors.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook gives a step-by-step method for orthogonally diagonalizing a real symmetric matrix. Students compute eigenvalues, build orthonormal eigenspace bases, and assemble P and D satisfying A=PDPᵀ.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook presents equivalent tests for an orthogonal matrix, including AᵀA=I, A⁻¹=Aᵀ, and orthonormal rows or columns. Exercises apply these criteria to symbolic and numerical matrices.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook connects invertible matrices with uniquely solvable systems Ax=b. Students compare Gaussian elimination with the inverse-matrix formula x=A⁻¹b and practice both approaches.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook studies when a matrix admits an LU decomposition and constructs the lower- and upper-triangular factors from elimination operations. Exercises guide students through the decomposition algorithm and its limitations.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook introduces matrix addition, scalar multiplication, and transposition, including their dimension requirements and algebraic properties. Exercises develop fluency with these basic matrix operations.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook develops invertibility criteria using homogeneous systems, pivots, and row reduction. It also reviews the augmented-matrix algorithm for computing an inverse and applies the criteria in exercises.