Mathematical Notation as Learnable Language

To utilize mathematical notation fully you need to interpret it. To read it fluently, you must map symbols to concrete lenses, e.g. computational, visual, algebraic, or descriptive.

Example: Bilinear Map

Let

$$f:R^2\times R^2\to R$$

be defined by

$$f\left(\right(x_1,x_2),(y_1,y_2\left)\right)=x_1{y}_1+2x_2{y}_2.$$

Interpretations:

1. Computational

   Substitute specific vectors and check results. If $v=(3,4)$, then

   $$f\left(\right(x_1,x_2),v)=3x_1+8x_2,$$

   Through this symbolic computation we can see how the expression depends on $x$. Perform such computations until you get a feel for the "shape" of the functions behavior.

2. Visual

   For each fixed $v$, the function $u\mapsto f(u,v)$ is represented by a hyperplane in $R^2\times R$. We can imagine walking on the hyperplane. This obviously always walks on a line, therefore it's linear.

3. Symbolic manipulation

   Verify algebraically:

   $$f(au+bw,v)=(a{u}_1+b{w}_1)v_1+2(a{u}_2+b{w}_2)v_2=af(u,v)+bf(w,v).$$

   This establishes linearity by direct algebraic manipulation. You understand what properties exist by showing them algebraically.

4. Descriptive

   What it means to be a bilinear map is, that if you hold the second argument fixed, and vary the second, you have a linear function. Same if holding the first fixed and varying the second.

   You want to capture the intuition in natural language.

Mathematical language is a language that you need to learn like any other. Often people get stuck by trying to use symbolic manipulation too much. Because mathematical language is so precise, it makes it easy to interpret it in many different while still being able to check if your interpretation captures the core.