Rank Theorem: Linear Applications

This lecture covers the Rank Theorem in linear algebra, stating that for a linear application T from a finite-dimensional vector space V to another vector space W, the image of T has finite dimension and dim V equals the sum of dim ker(T) and dim im(T). The proof involves showing that if V is of finite dimension, then ker(T) is also of finite dimension, and a base of V can be used to generate a base of im(T). The lecture also discusses linear independence, demonstrating that certain vectors are linearly independent if they form a base of V. The concept is illustrated through examples and proofs.