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We work with fields of numbers, such as $\Bbb Q$, the field of rational numbers, $\Bbb R$, the field of real numbers, and $\Bbb C$, the field of complex numbers. What is a field? It's a set in which two invertible operations - addition and multiplication - interact. Elementary algebra is simply the study of that interaction.

What's a linear function defined on one of these fields? It's a function that is compatible with the two operations. If $f(x+y)=f(x)+f(y)$, and $f(cx)=cf(x)$, then the whole domain, before and after applying $f$, is structurally preserved. (That's as long as $f$ is invertible; I'm glossing over some details.) Essentially, such a function is simply taking the field and scaling it, possibly flipping it around as well. In the complex field, the picture is a little more.... complex, but fundamentally the same.

The most intuitive vector spaces - finite dimensional ones over our familiar fields - are basically just multiple copies of the base field, set at "right angles" to each other. Invertible linear functions now just scale, reflect, rotate and shear this basic picture, but they preserve the algebraic structure of the space.

Now, we often work with transformations that do more complicated things that this, but if they are smooth transformations, then they "look like" linear transformations when you "zoom in" at any point. To analyze something complicated, you have to simplify it in some way, and a good way to simplify working with some weird non-linear transformation is to describe and study the linear transformations that it "looks like" up close.

This is why we see linear problems arise so frequently. Some situations are modeled by linear transformations, and that's great. However, even situations modeled by non-linear transformations are often approximated with appropriate linear maps. The first and roughest way to approximate a function is with a constant, but we don't get a lot of mileage out of that. The next fancier approach is the approximate with a linear function at each point, and we do get a lot of mileage out of that. If you want to do better, you can use a quadratic approximation. These are great for describing, for instance, critical points of multi-variable functions. Even the quadratic description, however, uses tools from linear algebra.

Edit: I've thought more about this, and I think I can speak further to your question, from comments, "why does the property of linearity make linear functions so "rigid"?"

Consider restricting a linear function on $\Bbb R$ to the integers. The integers are a nice, evenly spaced, discrete subset of $\Bbb R$. After applying a linear map, their image is still a nice, evenly spaced, discrete subset of $\Bbb R$. Take all the points with integer coordinates in $\Bbb R^2$ or $\Bbb R^3$, and the same thing is true. You start with evenly spaced points all in straight lines, and after applying a linear map, you still have evenly spaced points, all in straight lines. Linear maps preserve lattices, in a sense, and that's precisely because they preserve addition and scalar multiplication. Keeping evenly spaced things evenly spaced, and keeping straight lines straight, seems to be a pretty good description of "rigidity".

Does that help at all?