Data Science and Machine Learning (Theory and Projects) A to Z - Mathematical Foundation: Basis and Dimensions

The video tutorial introduces key concepts in dimensionality reduction, focusing on basis and dimensions within vector spaces. It explains the operation of span, illustrating how a set's span forms another set containing all linear combinations. The tutorial further defines basis as an independent set that spans a vector space, providing examples to clarify the concept. Finally, it discusses dimensions, explaining that the number of basis vectors determines a vector space's dimensions, and cautions against relying solely on coordinate count for dimension determination.

What is the concept of span in relation to a set?

What is the relationship between basis and dimensions?

How can the span of a set be defined?

What is the significance of linear combinations in defining span?

What theorem relates vector spaces to spans of sets?

Explain what a basis is in the context of vector spaces.

Can you provide an example of a basis for a vector space?

How does one determine if a set of vectors is independent?

What defines the dimension of a vector space?

Why might counting coordinates not accurately reflect the dimension of a space?