Linear Applications: Kernel

This lecture covers the concept of the kernel of a linear application between vector spaces. The instructor defines the kernel as the set of vectors that are mapped to the zero vector. Examples are provided to illustrate the concept, including the projection onto a plane and derivative operator. The lecture also discusses properties of the kernel, such as being a subspace of the domain space and its relation to the injectivity of the linear application. A detailed proof is given to show that the kernel is non-empty if the linear application is not injective. The lecture concludes with an example where students are tasked with determining the injectivity of a given linear application and finding its kernel.