Why study linear algebra?

Linear algebra is vital in multiple areas of science in general. Because linear equations are so easy to solve, practically every area of modern science contains models where equations are approximated by linear equations (using Taylor expansion arguments) and solving for the system helps the theory develop. Beginning to make a list wouldn't even be relevant ; you and I have no idea how people abuse of the power of linear algebra to approximate solutions to equations. Since in most cases, solving equations is a synonym of solving a practical problem, this can be VERY useful. Just for this reason, linear algebra has a reason to exist, and it is enough reason for any scientific to know linear algebra.

More specifically, in mathematics, linear algebra has, of course, its use in abstract algebra ; vector spaces arise in many different areas of algebra such as group theory, ring theory, module theory, representation theory, Galois theory, and much more. Understanding the tools of linear algebra gives one the ability to understand those theories better, and some theorems of linear algebra require also an understanding of those theories ; they are linked in many different intrinsic ways.

Outside of algebra, a big part of analysis, called functional analysis, is actually the infinite-dimensional version of linear algebra. In infinite dimension, most of the finite-dimension theorems break down in a very interesting way ; some of our intuition is preserved, but most of it breaks down. Of course, none of the algebraic intuition goes away, but most of the analytic part does ; closed balls are never compact, norms are not always equivalent, and the structure of the space changes a lot depending on the norm you use. Hence even for someone studying analysis, understanding linear algebra is vital.

In other words, if you wanna start thinking, learn how to think straight (linear) first. =)

Hope that helps,

Having studied Engineering, I can tell you that Linear Algebra is fundamental and an extremely powerful tool in every single discipline of Engineering.

If you are reading this and considering learning linear algebra then I will first issue you with a warning: Linear algebra is mighty stuff. You should be both manically excited and scared by the awesome power it will give you!!!!!!

In the abstract, it allows you to manipulate and understand whole systems of equations with huge numbers of dimensions/variables on paper without any fuss, and solve them computationally. Here are some of the real-world relationships that are governed by linear equations and some of its applications:

Load and displacements in structures
Compatability in structures
Finite element analysis (has Mechanical, Electrical, and Thermodynamic applications)
Stress and strain in more than 1-D
Mechanical vibrations
Current and voltage in LCR circuits
Small signals in nonlinear circuits = amplifiers
Flow in a network of pipes
Control theory (governs how state space systems evolve over time, discrete and continuous)
Control theory (Optimal controller can be found using simple linear algebra)
Control theory (Model Predictive control is heavily reliant on linear algebra)
Computer vision (Used to calibrate camera, stitch together stereo images)
Machine learning (Support Vector Machine)
Machine learning (Principal component analysis)
Lots of optimization techniques rely on linear algebra as soon as the dimensionality starts to increase.
Fit an arbitrary polynomial to some data.

Arbitrarily large problems of the types listed above can be converted into simple matrix equations, and most of those equations are of the form A x = b. Nearly all other problems are of the form A x = λ x. Yep, you read that right! Nearly all engineering problems, no matter how huge, can be reduced to one of two equations!

Linear algebra is so powerful that it also deals with small deviations in lots of non-linear systems! A typical engineering way to deal with a non-linear system might be to linearize it, then use Linear Algebra to understand it!

TL;DR version

Linear algebra is your ticket to multidimensional space. Study it if you are into economics, computer graphics, physics, chemistry, statistics or anything quantitative (in today's world, that's everything).

Meaning of "Linear" and why it is "Easy"

Since you are asking the question, perhaps you would benefit from a discussion of what "linear" means, and why it is "easy", as mentioned in some answers above.

The word "linear" is to be understood as in a linear function (a line) in calculus, such as

$$g(x) = mx + n,$$
where $m$, $n$, and $x$ are numbers.

You will know that in calculus, the Taylor expansion of a function $f$ is

$$f(x) = f(x₀) + f'(x₀)(x - x₀) + ½ f''(x₀) (x - x₀)^2 + \dots$$

If you use only the first two terms of the Taylor expansion of $f$, you have a linear approximation to your function $f$:

$$f(x) ≈ f(x₀) + f'(x₀)(x - x₀) = f'(x₀)x + (f(x₀) - f'(x₀)x₀)$$

If you set

$$m = f'(x₀), n = f(x₀) - f'(x₀)x₀,$$

you can see that this is a linear function just like $g(x)$. This approximation works reasonably well near $x₀$ (think of the $\sin x$ function, for example). It is the simplest approximation to $f$ beyond saying "near $x₀$, it's probably something like $f(x₀)$". That's why the linear approximation is called "easy".

If you want to do slightly better, use the first two terms of the Taylor expansion of $f$. You get a quadratic approximation to $f$.

How Linear Algebra Comes In

Now what do we do if we have a function that depends on several variables (or, which is the same, a variable $x$ that is a vector containing these variables as components)? This is called multivariate calculus (because, multiple variables).

In that case, $f'(x₀)$ becomes a vector that has to be multiplied (how?) with the vector $(x - x₀)$. $m$ and $n$ become vectors, too.

$f"(x₀)$ becomes a matrix (actually, a bilinear tensor) that has to be multiplied twice with the vector $(x - x₀)$. (How?)

As you have seen, the linear and quadratic approximations are the easiest ways to think about complicated functions $f$.

Linear algebra deals with vectors, matrices, and tensors, and how to operate them with each other so that everything makes sense and is useful.

In other words, linear algebra is your ticket to multidimensional space, such as the three-dimensional space we inhabit (e.g. computer graphics, physics) or the space of many variables a function depends on (economics, more physics).

Examples

Economics example: Price (our function $f$) depends on supply and demand (the two components of the two-dimensional vector $x$). If you do economics, you do linear algebra.
Politics example: If more roads are built, capacity goes up (good for business) but quality of life goes down. People move away, which results in more traffic, which results in less excess capacity, which is bad for business. Many variables and complex connections lead to multivariate calculus. Sociologists and politics majors ought to know linear algebra to think about problems like these.
Physics example: If you don't watch a moving car from the outside but sit in the car, you are changing your frame of reference. This corresponds to a change of basis in linear algebra, which naturally pops out the centrifugal force (and the Coriolis force) from your force expression.

For the mathematician

Linear algebra is of course a rich field in its own right but I wanted to write a motivating explanation to aspiring students who do not yet know what it is and how it is relevant to their lives.

Patrick Da Silva gives a good answer, however I will expand. Well worth looking at his post when (or if) you do get into Linear Algebra and come back to this post a year later or so with some knowledge under your belt, as you can easily extend your studies for a couple of years by what he stated.

Your beginning motivation to study linear algebra is to put together what you initially learn in (mathematically) geometry and calculus and in (education) high school, college, or even first year or second year of university.

Imagine there's some "room" where everything you played around with in A-Levels or high-school was contained; $\sin(x), \cos(x), ln(x)$ were there, a definition of a field and some set theory.

Now suppose we want to understand this room. Suppose instead of looking at each little "person" in the room, we know that if they are in that room, they behave under a specific way.

Linear Algebra (and Algebra in general) allows us to classify and understand many objects, situations, spaces, to some basic context. At the a high undergraduate level, a remarkable theorem in Hilbert Spaces allows you to identify any separable Hilbert space to some specific, well-known Hilbert Space. This is just the work of the last paragraph in application.

Therefore, if you want structure, you need algebra. Linear algebra specifically so you can understand the basic structure.

One application of Linear Algebra is in the use of eigenvalues.

We can understand some specific systems; weather predictions, behaviour of people, game situations, etc, and the long run behaviour (as it approaches $\infty$ so as to speak) by instead of looking at some function $f(x)$, we look at some real value $\lambda$ such that $f(x)=\lambda x$ at some meaningful $\lambda$.

Linear Algebra allows us to start understanding basic linear systems with use of matrices and vectors.

Lastly, for a purely computational reason, Linear Algebra gives you many tools for proving many key theorems. The infinite dimensional Pythagorean theorem, Cauchy-Schwarz inequality, Bessel Inequality, etc, are frequently used in many areas of mathematics to prove key theorems. This is good motivation for any subject.