Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook studies dimensions of subspaces, intersections, and sums in finite-dimensional vector spaces. Exercises apply dimension bounds and the dimension formula for W+U to concrete families.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook gives a row-reduction method for extracting a basis of a subspace from a finite spanning set and extending it to a basis of the ambient space. Exercises apply the method through coordinate matrices and pivot columns.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook defines a basis as a linearly independent spanning family and introduces finite-dimensional vector spaces. Students practice deciding whether collections form bases and relating coordinates to uniqueness of representation.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook defines dimension as the number of vectors in a basis and gives basis criteria for a space whose dimension is known. Exercises use the size of independent or spanning families to conclude that they form bases.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook fragment presents the definition of linearly dependent and independent vector families. It highlights the immediate dependence of any family containing the zero vector and supports the chapter's interactive courseware.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook presents the rank-nullity theorem for linear maps and derives dimension constraints for injective and surjective maps. Exercises compute kernels and images to decide injectivity and surjectivity.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook applies rank-nullity to linear maps between finite-dimensional spaces of equal dimension. Students use injectivity or surjectivity to prove bijectivity and identify when two vector spaces are isomorphic.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook introduces the Euclidean dot product, vector norm, and angle in R² and extends the ideas to R³ and Rⁿ. Exercises compute lengths, dot products, and angles while interpreting their geometry.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook defines orthogonal and orthonormal families and bases. It derives coordinate formulas in an orthogonal basis and shows how an orthonormal basis preserves the usual dot product of coordinate vectors.

Simone Deparis

MATH-111

Python3

Interactive textbook

English

This notebook defines orthogonal complements and decomposes a Euclidean space as W⊕W⊥. It derives projection formulas from orthonormal bases and applies them to subspaces through interactive Gram-Schmidt exercises.