Natural Blogarithms

Today, in class, I answered some homework questions from Section 1.4, including the following problems: [...]

In class today, we finished the section on partitioning matrices. We verified that that block matrices obey the same rules for matrix algebra. In particular, block matrix multiplication works as scalar multiplication as long as the dimensions are appropriate for the sub-matrices to be multiplied together. [...]

On Wednesday, last week, we completed our material over elementary matrices, using them to derive the inverse of a matrix. Upon proving that a matrix is non-singular (i.e., invertible) if and only if it is row equivalent to the identity, we noticed that the same row operations that change a matrix A into the identity will change the identity into the inverse of A.

I don’t think it is possible for me to cover a whole section of this linear algebra book in one class. Today, we started the section on Elementary Matrices and just a little ways in, I knew it was going to be a difficult section. [...]

It took three classes but I finally finished all of the section over Matrix Algebra. During today’s lecture we walked through the concepts of matrix inverse and then matrix transposes. Its amazing how long these lectures stretch out when you choose to demonstrate the concepts with examples involving matrix operations. As far as I can tell, they are all following along very closely, except when I start going off on a tangent and trying to draw connections between linear algebra and higher level mathematics. [..]

During Linear Algebra on Monday, I began class by answering homework questions. … Somehow the discussion of a applications of techniques we are learning came up. Thus far, we have basically covered how to use Gauss-Jordan elimination to solve linear systems. I pointed to examples from engineering (such as structural analysis of trusses) and computational fluid dynamics (discretization procedures to solve PDEs). I understand that these are a little outside the scope of this class but I chose them because of the individuals asking the question and based on their particular interests in mechanical engineering and aeronautics. [..]