## MATH 245 midterm

So the midterm for MATH 245 (Linear Algebra 2) is next week (Tuesday). The following topics have been covered in class, just as a quick overview. Everything we’ve done so far is in the setting of $\mathbb{R}^n$.

I’ve decided that (hopefully, time permitting) I’m going to type up a document which will colloquially overview all the stuff we’ve done so far. It will focus more on the important ideas behind the theory rather than all the details. I’m hoping to finish it by the end of Sunday, i.e. about 2 days before the midterm. I will just edit this post when it is done (also, it will be available on the front of dx7hymcxnpq.com).

• Affine spaces in $\mathbb{R}^n$, basic facts
• Affine span, affine independence, convex hulls
• Simplices
• Higher-dimensional generalizations of triangle centers to simplices: centroid, circumcenter, orthocenter
• Dot product, norms, and angles and their properties: Cauchy-Schwarz, the generalized law of cosines, polarization identity, etc.
• Orthogonal complements and orthogonal projections, reflections (used these concepts to talk about the circumcenter of a simplex)
• Best-fitting/least-squares polynomials and the Vandermonde matrix
• Generalized cross product in $\mathbb{R}^n$ and formal determinants
• $k$-volume of parallelotopes
• Relationships (particularly geometric, i.e. orthogonality) between different subspaces associated with a matrix and its transpose: nullspace/kernel, column space, row space, etc.