Are matrices more easier to work with than linear maps? I never liked doing computations with matrices. This was before I learnt that matrices are used to represent linear maps. Also that the definition of matrix multiplication stems from trying to find the matrix of the composition of two linear maps. Another property of a matrix of some linear map is that a matrix multiplied by a vector is the same as evaluating the linear map for that specific vector. What I don't understand is that why is the use of matrices in applications so emphasised. I could be wrong, but wouldn't it be easier to evaluate a function than to multiply a matrix with a vector. The only useful way I see matrices making computations easier is when one wants to find the inverse of a linear map that is hard to find just by the expression of the map. Even while learning about matrices for the first time, a lot of problems phrased in terms of matrices looked like they would be phrased better in terms of functions or just equations.
My question is that, how is it more useful to use matrices than it is to use linear maps?
 A: If you want to do concrete calculations with linear maps, then you are going to do calculations with matrices. If you're solving sets of linear equations with the addition method, then you are working with matrices. There may be other ways of organizing the numbers; it doesn't have to be a rectangular grid. But ultimately you can't really get away from matrices. There's a reason they are popular.
It is true that many general arguments in linear algebra are possible to make entirely without touching matrices. So they aren't needed for everything. And I also agree that many linear algebra courses and books focus a little too much on matrices and not quite enough on linear maps. But they are a necessary tool in many cases.

wouldn't it be easier to evaluate a function than to multiply a matrix with a vector.

Multiplying a matrix with a vector is evaluating a function. It is evaluating a certain linear map (given to you in the form of what it does to basis vectors, with the outputs expanded as linear combinations of basis vectors) on a certain vector (given to you as a linear combination of basis vectors).
A: Complementary remarks relatively to the other answer
Matrices are concise and easy way to define a linear maps and operations betwen linear maps. A linear map $f$ on a finite dimension space is defined by the image of a basis of that space by $f$. Indeed a linear map between two vector spaces $f:U \mapsto V$ can be represented as a matrix by letting:
$$A = \begin{bmatrix} f(e_1) \dots f(e_n) \end{bmatrix}$$
where $e_1,\dots,e_n$ is the canonical basis (but not necessarily).
Despite the purely mathematical developments, a 2D table is easy to implement and to work with on a computer. Matrices are the key point of any simulation software I can think of; weather forecast, wave propagation, solid mechanics, ... To be solved, non-linear problems are translated into a linear system of equations, which can be in turn be seen as matrices and vectors. For a "real world" problem, we are talking of systems with hundreds, thousands or even millions equations. Having such a concise way to write, to understand and to perform operations on linear maps is therefore critical. Matrices can also be found behind search algorithms (via stochastic matrices), trafic optimization (via linear programming), and so on ...
A: I also wanted to give a small input into my point of view on how one can think about the two (the other answers are awesome they just miss a keypoint for me).
In my experience it is similar to the different definitions of compactness ("every open cover has finite subcover", or in nice enough spaces, "bounded and complete"). In most computational cases I recommend working with matrices, but only if you really work with ONE concrete map. If you want to argue generally it usually is better to think of linear maps. This even applies to arguments about the rank, a lot of arguments are actually a lot clearer when one considers the rank of a matrix/morphism to be the dimension of the image.
All in all I would just see it as a tool, depending what you are looking at, one of the tools is often better suited, in this case at least for me: matrices for concrete concrete computations and linear maps for all the rest.
