Go through Section 2.4 of the Textbook. As you can see, we are not strictly following the order in the text. This is because this course has a companion lab, and it allows us to use class time in more effective ways. In class we will streamline the treatment of linear equations, and in the lab you will see linear equations in lot more detail. The text is encyclopedic. While going through the material, focus on the following amidst the myriad mechanics of matrix mulitplication:

• the ability to identify which linear combinations are happening
• in the subsection “Rows and Columns of \(AB\)”, note that each column of \(AB\) is \(A{\bf b}_i\), hence a linear combination of the columns of \(A\)
• similarly, each row of \(AB\) is a linear combination of the rows of \(B\).
• \(AB \ne BA\) in general for matrix multiplication. Make sure you remember this when you use the associative law: when you compute \(ABC\), you can do it either as \((AB)C\) or \(A(BC)\), but not as \((BA)C\) or \(A(CB)\) or any other permutations. In fact, some of the products \(BA\) or \(CB\) may not even be defined. Students often mess this around.
• Note in particular that for matrices \(A\) and a real number \(c\), \(Ac = cA\) multiplies every entry of \(A\) by the number \(c\). Therefore, \(A(cB) = (Ac)B\) and in this special case, it happens that \((Ac)B = cAB\). Read the prior note again to recall that this is special when \(c\) happens to be a real number, and not true if \(C\) were a matrix in general.

For now, you can downplay the subsection “Block Matrices and Block Multiplication”, as well as the Schur complement. We will come back to it later in the course.