MIT OCW 18.06SC Sections – Lectures & Reading:
- Exam
- Orthogonal Vectors & Subspaces
- Strang 4.1
- Projections onto Subspaces
- Strang 4.2
- Projection Matrices & Least Squares
- Strang 4.3
- Orthogonal Matrices & Gram Schmidt
- Section 4.4
Hours Spent ~ 60 hours
Calculus 3 Professor Leonard
- Lecture 11.1 & 11.2 – Introduction to Vectors & Vectors in 3d
- Thomas 13th edition 12.1 Vectors in the Plane
- Thomas 12.2 Vectors in Space
- Lecture 11.3 The Dot Product
- Thomas 12.3 – Dot Product
Hours Spent ~ 12 hours
I did it again – I neglected my blogging duties. I was sure I’d get back on track, writing once a week. Nevertheless, I procrastinated for four weeks. I think my routine is so routine, my days so very similar, that I lost track of time. If you asked me what I did last week, I could tell you “I studied math, I worked on my business, I went to the gym, etc.” However, I don’t remember a single defining feature about my week. I don’t ever meet people, so everything bleeds into one. And a week becomes a month.
Since I last wrote I took the first Linear Algebra exam. I didn’t know what the grading metric for the exam was, but I felt pretty good about my outcome. I was pretty solid on the tested concept. As is often the case in Linear Algebra, the most difficult part was interpreting what the questions really meant.
I’m now working in Unit 2 of the OCW course – Least Squares, Determinants, and Eigenvalues. Thus far the focus of the studies has been on the concept of orthogonality. The unit starts out kind of cryptic, with the introduction of the four subspaces and orthogonal complements. The idea of orthogonality is clear enough – think vectors whose dot products result in zero. But it quickly leaps off into the fog with the concepts of orthogonal subspaces and the combination of bases in subspaces. Questions hurt the brain – no computation, but rather “which subspace does this vector live in?”. I do my best to formulate some kind of mental picture of the interplay between the subspaces, but I’m sure I fall short. The left-nullspace? C’mon!
I’ve decided that I must accept a degree of uncertainty with respect to Linear Algebra. I can’t fully grasp what’s going on and keep moving at a steady pace. I get the gist (or at least I think I get the gist) of what’s going on, then I proceed, and further topics reinforce what I learned, so my brain catches up while at the same time bobbing around in the pool of uncertainty.
What’s great about this unit is that the why of orthogonality and the subspaces makes sense as we move forward into Projections and Least Square Approximations. Up until this point, we’ve been working with Ax = b equations that always have solutions. Naturally, many/most equations don’t have a clear-cut solution. We’re able to apply orthogonality to unknown vectors onto vectors for which we have clear equations. I could see immediately where this would have strong applications, by fitting trends, be they lines or matrices, that aren’t exactly linearly defined, into lines or matrices that are defined, and charting corresponding margins of error.
This whole business takes some getting used to, as projection matrices calculations often look like this: Over time we get used to symbols, and then they continue to stack on and make sense. What an amazing phenomena it is to learn and build whole new models of mathematical understanding.
We continue to run with this idea as we construct orthonormal vectors, i.e. those which are orthogonal to all others in their column space, and are unit vectors when multiplied by themselves. Finally, the point is reached where we can take linearly independent vectors and transform them into Q matrices of orthonormal vectors and upper-triangular R matrices using the Gram-Schmidt process. Imagine being Gram or Schmidt. “I was the guy that added Q to the mathematical canon.”
Linear Algebra is mind-bending, and I love it. Each new section reveals a little more of what’s going on underneath, like a beautiful lady slowly hiking up her dress to reveal sublime thigh, while always leaving me wondering what lies beyond the fold. I look forward to see what she let’s me peak at next.
As I’ve recently freed up some time from the Coder Pro ball-and-chain, I’ve added on Calculus 3 to my daily tasks. I’m back with Professor Leonard! I want to have my math in line to be sufficiently prepared to start the DEDP Micromasters in September. In order for that to happen, I need to knock out Linear Algebra, Calculus 3, Probability Theory, and Statistics.
I feel like Calculus works a different part of my brain than Linear Algebra, so adding it on will be feasible. The first few sections have all felt very familiar, as they have dealt with vectors, their representation, and dot product. I’ve been working with vectors and dot products for the last few months, so I’ve been able to move through the lectures and homework without too much friction. It’s nice to be back to good ol’ Professor Leonard to laughing during lectures. Yes, we’re back!