Even Even More Problems For Dimensional Analysis: Fill & Download for Free

In seventh grade, in Physical Science, we started learning Dimensional Analysis.Dimensional Analysis is the process by which (usually physical) units are translated into other units, by lining up conversion formulae such that the top and bottom units can be cancelled. For example:This is a simple example; we start with two “dozen eggs”, and we want to understand how many “eggs” that equals. By lining up the 12 eggs / dozen eggs translation, we can cancel out the “dozen egg” terms, multiply out the numerator and denominator, and get our answer.I, as a very stubborn and clever-enough seventh grader, did not need to line up fiddly terms; I could power through this conversion by just “thinking” about what to multiply and divide by. In this case, you multiply by 12, obviously. Two “hops” was also simple enough to hold in my head:[source]. I just kinda figured out that you divided by 3600 and multiplied it up to get meters per second.Since every example we covered in class was easy enough to reason through, I steadfastly refused to learn the fiddly process of lining up and cancelling out like-units. It was an unnecessary waste of time, in my mind. I made it through homework and quizzes without learning the method, missing a few points for not showing my work, but getting answers right.Well. It turns out DA was a tool, which we would then use as a foundation for everything else. So a few months later, we started using DA, not just learning it. The problems got much, much more complex:Complex enough that no, I could not just “reason” my way through the conversion formula in one hop. I hit the complex problems, and instead of learning the actual lessons, I got stuck on the “easy” part — the unit conversions.Unfortunately for me, the rest of the class wasn’t learning DA anymore. They were using it. I flailed even getting started on homework problems where others were able to jump immediately to the meat of the problem.As a reasonably clever, stubborn 7th grader, could I admit that I had messed up and needed remedial help on the basics? Of course not. I floundered for months, refusing to admit I had screwed myself over by intuiting answers and not learning the fundamentals.I do sympathize with people who hate showing work, and I agree that showing work on everything is a waste of time which kills the fun in problem-solving. But it would have saved me considerable frustration had I been forced to work through the basics at least a couple times, to prove that I was learning tools and not just brute-forcing an answer.(I do not mean to imply I am some kind of math prodigy. I am not. I was simply capable enough to do easy seventh-grade math in my head instead of from first principles. That is all.)

You might want to look up and study the Buckingham π theorem. It may give you more insight into why dimensional analysis is useful, and it formalizes what Barak Shoshany and Joshua Engel are saying in an interesting way.The real value of dimensional analysis I think is actually that given some physical equation among a set of n variables, and given that the n variables can be expressed in terms of k dimensionful units, then it is always the case that the original equation can be reduced to a set of equations among p dimensionless variables, where p=n-k.With a clever choice of the dimensionless variables, a physical problem can sometimes be greatly simplified, and a lot of physical insight can be gained that way.And this can always be attempted, even when the form of the physical equation isn't yet known but has to be guessed at.The formal proof is worth studying, it depends on the rank-nullity theorem for linear transformations on vector spaces which are cleverly defined on spaces of the dimensionless and the dimensionful variables.Looking at the formal proof may help demystify the process of dimensional analysis for you.Or maybe not ;)

If you work as an entry level data scientist at a startup (as you mention in the question description), chances are you don't really need much linear algebra knowledge, as familiarity with scripting languages (python, R), tools like scikit-learn, pandas and database technologies are often more important than understanding the maths behind machine learning.That said, the Matrix cookbook is a useful resource: Matrix CookbookIt really depends on what types of problems you will come across.Singular value decomposition and particularly understanding it's relation to principal components analysis is probably the single most useful concept to know. See my answer to Is there a relationship between QR decomposition, ICA and PCA? and references therein.Understanding projections and least squares type optimisation problems is useful for dimensionality reduction, clustering and even regression.At a more advanced level, reproducing kernel Hilbert spaces are often used in machine learning (Gaussian process, support vector machine, kernel PCA) You can learn about these from Learning with Kernels: Support Vector Machines, Regularization, Optimization and Beyond by Bernhard Schoelkopf