Welcome to Week 6.
This week, we will build on our discussion of linear independence and spanning sets from last week, and introduce the fundamental concepts of basis and dimension. In Lecture 11, we will prove the Steintiz exchange lemma, which states that in every vector space, the number of elements in a linearly independent set cannot exceed the number in any spanning set. We will deduce that all bases of a finite dimensional vector space have the same number of elements.
Here is the recorded version of Lecture 11.
In Lecture 12, we will develop the concepts of basis and dimension a bit more, and look at some relationships between different bases. We will also define some important spaces related to matrices and to linear transformations.
Here is the recorded version of Lecture 12.