Chapter 3.3 works with the pivots of a matrix. Some content from Chapter 3.3 is defined only in Chapter 3.5, and most of Chapter 3.3 is better understood in the context of Chapter 3.5.

The very first definition in Chapter 3.3 says “the rank of \(A\) is the number of pivots”. The narrow interpretation is that this is a definition that is based on how we define Gaussian elimination, but the reality, as you learn later is that it is not. It does not matter how you reorder the columns, whether you modify Gaussian elimination (in a sane way) or more. Chapter 3.3 states: “The rank is the dimension of the column space of \(A\)”, but at this point in the text, the term/concept “dimension” is not yet defined, and you have to wait till Chapter 3.5 to know what is meant. Also, as we showed in class, a minor miracle is that the row space of \(A\) has the same dimension.

The development in class goes from Chapter 3.5 \(\to\) Chapter 3.3. A lot of insights in Chapter 3.3 must be familiar to you even before you get here (for eg. Every free column is a combination of earlier pivot columns). In fact, you also know how to get the exact linear combination of the prior columns that yields that particular free column (from the rref).