| HN Mirror

> What's a linear transformation? > I get that it's f(x + y) = f(x) + f(y) and f(cx) = c * f(x)... > but what does that really mean?

That's a really good question!

Essentially, when some function (synonym for transformation) is linear, what this tells you is that it has "linear structure," which in turns out to be a very useful property to know about that function.

You can combine the two facts you mentioned above to obtain the equation

   f(a*x + b*y) = a*f(x) + b*f(y)

for which the interpretation is that the linear combination of inputs ax + by is transformed into the SAME linear combination of outputs af(x) + bf(y).

Suppose now that the input space of f can be characterized by some finite set of "directions" e.g. the x- and y-directions in case f is a 2D transformation, or perhaps the x-, y-, and z-directions if f is a 3D transformation. If f is a 3D transformation, using the linear property of f, it is possible to completely understand what f does by "probing" it with three "test inputs," one along each direction. Just input x, y, and z, and record the three output f(x), f(y), and f(z). Since you know f is linear, this probing with three vectors is enough to determine the output of f for any other input ax + by + cz --- the output will be af(x) + bf(y) + cf(z).

See the same explanations as above but in more details here: https://minireference.com/static/excerpts/noBSguide2LA_previ...

So why is this important? Well this "probing with a few input directions" turns out to be really useful. Basically, if f is non-linear, it means it super complicated and would be no simple way to describe what its outputs are for different inputs, but if it is linear then the "probing procedure" works. Furthermore, since both the inputs and outputs have the form of a linear combination (a constant times something + another constant times another thing + a third constant times a third thing), you can arrange these "things" into an array called a matrix and define an operation called "matrix multiplication" which performs the constant-times-something operation of outputs, when you give the constants as an array (vector) of inputs.

In summary, linear algebra is a bunch of machinery for expressing various transformations in terms of vectors and matrices in order to help with modelling various real-world phenomena ranging from computer graphics, biology, chemistry, graphs, crypto, etc. Even ML ;)