Linear algebra is in many ways the bread and butter of mathematics. It plays a central role in almost every mathematical subject (pure or applied) you will encounter from here on in. There are so many applications that a solid understanding may just change the way you look at the world!

Let us put aside, for a second, what exactly linear algebra is about, except to say that it studies objects which are "linear." What might be some linear objects? Your first guess might be a straight line. Maybe if you dare to think in higher dimensions, you might imagine a 2-dimensional plane sitting inside of three-dimensional space. Or if you dare even further, you might arrive at the notion of a k-dimensional plane in some n-dimensional space (for k ≤ n). The power of linear algebra then relies on three key points.

(1) Linear objects are "rigid" enough that they come with a lot of structure.

In mathematics, when we study objects, we need a starting point. If we start with objects that are too loose or flexible, then there is less that you can say about them in general. On the other hand, linear algebra is centered upon a principle of rigidity. Think of it this way: it is easier to make an interesting statement about lines than it is about curves. For example, there is a unique line through any two points. This is certainly not the case for curves in general! By restricting to more rigid objects, we uncover more interesting mathematics. This first point may be viewed as the fundamental tenet of this class. We will spend the semester uncovering the structure of linear objects!

(2) Linear objects describe many naturally occurring phenomena.

Solving systems of linear equations is itself ubiquitous, and something which has occurred repeatedly in almost every math course you have taken in college (calculus, differential equations, etc.). When you find a least-squares approximation to a collection of data points, you end up having to solve a system of linear equations. The motion of objects in an inertial frame is linear unless acted upon by some force. Special relativity is essentially just a thorough analysis of linear operators which "preserve" a special matrix. Markov chains describe certain random discrete processes and are described in the language of linear algebra.

(3) Smooth non-linear phenomena are approximately linear.

Here, in my opinion, lies the greatest reason to study linear algebra. Let us think about calculus. When you take the derivative of a function f, you obtain a new function f', the value of which is giving the slope of a tangent line. In other words, even though the function is non-linear, when x is close to p, we have the approximation f(x) - f(p) ≈ f'(p)(x-p). The field of analysis (in which one rigorously defines the concepts in (multivariable) calculus) is in fact founded on this concept: show that properties of linear functions also hold for non-linear functions by using these approximations. The main technical detail in analysis is understanding what the sign "≈" means in a rigorous fashion. The underlying linear algebra (which we may conflate with understanding linear functions), on the other hand, is taken for granted because it is so well understood!

This idea is more powerful than you may think. For example, many physical phenomena are modelled using differential equations, so suppose I give you some differential equation modelling the real world, and I give you some input parameters, and ask you to simulate what will happen in the future. We often input such systems into computers to see the result, and any computer ever is using ideas from numerical analysis in order to approximate the solutions to these differential equations. For example, you may be familiar with the Euler method for numerically solving a differential equation. Notice that the figure on the Wikipedia page is specifically using straight lines. Nearly every physical simulation which has been performed on a computer essentially works via a souped-up version of this idea.

From the pure mathematics perspective, one often studies abstract objects by attempting to view them through the lens of linear algebra. For example, if you have taken abstract algebra, you may have seen the definition of a group. Regardless of what the word means, one often studies groups by means of a representation, which is a manner of mapping the group into the world of linear algebra. The power of this approach of attempting to study groups via their underlying linear algebra has many consequences in mathematics and also in modern physics, in which particles are secretly (smooth) representations of certain special groups!

No matter what you end up doing in the future, with just a little effort, it's not too hard to find linear algebra around you.