# Category Archives for **Math**

## Dual spaces, transposes, and adjoints

What’s the dual space of a vector space? Let $V$ be an $n$-dimensional vector space, with basis $$ B = \{ v_1, \ldots, v_n \}. $$ Now consider an arbitrary linear function $f$ that takes a vector $v \in V$ … Continue reading

## Proof of separation of variables

Let’s say we have a differential equation: $$f(y) \dod{y}{x} = g(x)$$ Separation of variables says that we can simply “split” the derivative $\dod{y}{x}$ and write $$f(y) \dif{y} = g(x) \dif{x}$$ Then, we can integrate both sides: $$\begin{gather*} \int f(y) \dif{y} … Continue reading

## Why does algebra work?

When I was in middle school, my teachers taught me the mechanics of algebra — try to isolate to variable by adding, subtracting, multiplying, and dividing both sides. And then in later years, my teachers expanded that toolbox to include … Continue reading

## Explanation of the Dirichlet function

My calculus teacher gave an example of a function that was discontinuous everywhere, the Dirichlet function. It indicates whether a number is rational or irrational: $$D(x) = \begin{cases}1 & \text{ if } x \in \mathbb{Q} \\ 0 & \text{ otherwise }\end{cases}$$ … Continue reading

## Minimum and maximum of two functions

Let’s say you want a function that outputs the smaller of two functions along its domain. It turns out that the expression for that function is this: $$\operatorname{min}(f(x),\ g(x)) = \frac{f(x) + g(x) – |f(x) – g(x)|}{2}$$ For example, take … Continue reading