In this post, we will compute a classical determinant called the Vandermonde determinant. Though the computation of this determinant looks intractable at first, there turns out to be a beautiful formula for it, with a very neat proof. Somewhat surprisingly, the matrices involved and this result are related to the notion of Lagrange interpolating polynomials. … Continue reading Vandermonde matrix
Asymptotic comparison – III
This is the third post regarding the notion of asymptotic comparison. The first one dealt with equivalents, the rigorous way of saying "these two things look alike", while the second post was about little o, meaning "this thing is much smaller than this other one". Naturally, where there is a little o, there should be … Continue reading Asymptotic comparison – III
Polynomials
Polynomials are probably the most usual types of functions out there, because they do not need any heavy machinery to be introduced: all one needs to know is how to add and multiply numbers. This alone justifies a discussion of polynomials. However, seeing polynomials as merely "simple functions" is reductive, and indeed, a more abstract … Continue reading Polynomials
Asymptotic comparison – II
This is the second post regarding the notion of asymptotic comparison. The first one dealt with equivalents, the rigorous way of saying "these two things look alike". This post is about little o, which describes what we mean by "this thing is much smaller than this other one". Yes, the "o" is the letter o, … Continue reading Asymptotic comparison – II
Functions
What is a function? It would be tempting to answer this question by writing a function. It probably involves a variable, a couple exponents, some additions and products; maybe, to make the heart content, top this up with a bevy of trigonometric functions, logarithms, and other exponentials. What do we get? Well, some kind of … Continue reading Functions
Asymptotic comparison – I
In the coming posts, I want to discuss the notions of asymptotic comparison, which is essentially the mathy way of saying "these two things look alike" or "this one is much larger than this other one". The point, as with many new notions, is to simplify. Most of the expressions / formulas / functions that … Continue reading Asymptotic comparison – I
Matrix multiplication
How do you multiply matrices? I realized that the technique that I learned back in the days is not as common as I thought. I often see my students struggling to perform simple matrix multiplications, and I would argue that part of the reason is that they write them side-by-side. This is also what I … Continue reading Matrix multiplication
L’Hospital rule
I had never heard of L'Hospital rule before I had to teach it, one fateful morning of 2012 at the University of Toronto. The class was your standard Calculus 1 class, where L'Hospital rule features prominently as your one-size-fits-all trick to compute limits. "How come I never heard of this?", I thought. That's when I … Continue reading L’Hospital rule
Let epsilon be positive 