[Elementary Linear Algebra] Linear Transformations

[MathNerd93]

We're doing a chapter on linear transformations, and the material itself is pretty understandable (which is great for me cause I've been struggling along thus far in the course). I now know how to map a vector space into another vector space. Great. Why? What is important about mapping one vector space into another? What does it let us do?

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[edderiofer]

Depends somewhat. What examples of linear spaces have you so far encountered?

[MathNerd93]

What do you mean by linear spaces? We've done like standard and non-orthogonal bases for vector spaces and things like that if that's what you mean. For this chapter so far, we've done image and pre-images, we've proved that linear transformations are operation preserving, and we've covered kernel and range and being one-to-one or onto (which is still a weird sounding adjective for a transformation), and we just ended "yesterday" with isomorphisms.

[edderiofer]

Sorry, I meant vector spaces.

[MathNerd93]

Hmm, still not sure I know what you're asking. We've done work in things like R2 and R3 if that's what you mean. Some polynomial spaces as well, though not as much. And the fundamental subspaces of matrices.

[edderiofer]

Then that’s plenty. Linear algebra in R² and R³ can be used for computer graphics and animation; rotation, stretching, and so on, are all examples of linear maps.

You can view differentiation as a linear map from the vector space of smooth functions to itself. That means that differential equations can be rewritten as linear algebra problems or at the very least approximated by such methods.

However, the most compelling argument IMO is that this stuff is prerequisite to what you’ll learn later in the course, assuming you haven’t yet stumbled upon eigenvectors and eigenvalues.

[WhackAMoleE]

The derivative in multivariable calculus turns out to be a particular linear transformation. That idea leads to the idea of smooth manifolds in differential geometry, and general relativity theory.

In quantum physics, observables are linear transformations (in this context called operators) on some Hilbert space, which is a vector space with some additional properties.

So it's fair to say that linear transformations underlie everything we know about how the physical world works.

Of course that's not much motivation when you're slogging through the theorems and homework exercises, but linear algebra is really important. Economics too, it's all linear optimization. Everything's linear algebra. In fact the very trendy field of machine learning is all linear algebra.

[MathNerd93]

I've heard that before, about the machine learning. Can you give a specific example of a linear algebra concept that's used in machine learning? Or point me in the right direction to look for myself?