Welcome to Week 5.
This week, we will discuss probably the most important ideas in linear algebra: spanning sets, linear independence, bases and dimension. In Lecture 9, we will consider the linear span of any subset of a vector space, and see what it means for a subset to be a spanning set for the whole space. We will also consider when the deletion of an element from a set changes its linear span, and when it does not.
Here is the recorded version of Lecture 9.
In Lecture 10, we will discuss the concept of linear independence and prove one of our most important theorems about vector spaces: that the number of elements in a linearly independent set cannot exceed the number of elements in any spanning set. From this point we will arrive at a clear statement of the meaning of the dimension of a vector space. Our work in this chapter is situated in the context of abstract vector spaces, but most of it admits quite concrete interpretations in terms of matrices.
Here is the recorded version of Lecture 10.