Vector Spaces

This chapter explores one of the fundamental topics in linear algebra. It starts by defining vector spaces, highlighting their importance as mathematical structures with essential qualities such as closure under addition and scalar multiplication. Subspaces are introduced as vector space subsets with their vector space features, followed by an in-depth analysis of linear dependence and independence of vectors, which are critical for constructing bases. The ideas of span and basis are emphasized as critical tools for understanding the structure of vector spaces, with dimension serving as a quantitative measure of their complexity. Finally, the chapter looks into vector space sums and the particular case of the direct sum, providing a more in-depth understanding of vector space composition.