Welcome to Week 6.
This week, we will build on our discussion of spanning sets from last week, and introduce the fundamental concepts of linear independence, basis and dimension. In Lecture 11, we will consider what it means for a subset of a vector space to be linearly independent. In Lecture 12, we will consider the relationship between spanning sets and linear independent sets, and prove the all-important Replacement Lemma, which states that the number of elements in a linearly independent set cannot exceed the number in any spanning set. We will deduce that all linearly independent spanning sets, known as bases, have the same number of elements.
Slides for this week’s lectures.
We are in Section 2.2 of the lecture notes this week.
Here is an old recorded version of Lecture 11.
Here is an old recorded version of Lecture 12.