Vector Spaces

Explore the fundamental concepts of vector spaces in linear algebra through 25 illustrative examples. Delve into subspaces, spans, nullspaces, linear independence, coordinates, and basis. Learn to determine whether a set forms a subspace, understand the properties of spans and nullspaces, and master the concept of linear independence. Investigate coordinate systems, including coordinate matrices and changes of coordinates. Examine the relationship between basis and dimension, and learn how to verify if a set of vectors constitutes a basis. Study the nullspace, column space, and rank of matrices, and explore infinite-dimensional vector spaces. Apply your knowledge to solve integration problems using coordinate changes in this comprehensive 3.5-hour course on vector spaces.

linear algebra vector space (25 examples).
Subspace.
Not subspace.
Span is a subspace.
Nullspace of a matrix.
Nullspace is a subspace.
Linear Independence.
Coordinates.
Coordinates matrix.
Change of coordinates.
Basis and dimension.
Check something is a basis.
Nullspace Column Space and Rank.
Infinite dimensions.
Integrate via change of coordinates.