Is linear algebra developed any further? I heard an opinion that linear algebra has ceased to develop 100 years ago because there's nothing else to "discover" in this branch of mathematics, and no scientific activity other than teaching occurs in this field.
Is this true?
 A: Linear algebra came to life again as a research area with the rise of numerical linear algebra. A lot of applied mathematicians work on developing algorithms to perform matrix computations efficiently. For example, iterative methods (such as multigrid methods) for solving large sparse linear systems have been investigated a lot in recent years. 
A: Linear algebra has different origins and has a paradoxical history of development (the concept of determinant for example, can be traced back to the middle of the 18th century, whereas, the concept of matrix was born among Hamilton, Cayley and the english school in the 1850s, even if Gauss had already worked on quadratic forms before. Then the German school, mainly after 1870 was the "flame holder" for half a century; and we are in the 1920s, a century ago, as you remarked. 
But, at that time, although many fundamental concepts of linear algebra were mastered (for example Hilbert spaces), it was not widely taught. It is by chance for example that Schrödinger attended lectures on matrices giving him the idea of using them in his new theories in the 1920s.
But at that time, many many other concepts were not born. 
@littleO has mentionned the huge revival of linear algebra with computer science ("US school" but this term is rarely used). The main factorizations, beginning by the most central one, the Singular Value Decomposition, but as well, Cholesky and QR decomposition.
the concept of conditionning, of ill posed problem, etc... and methods such as finite elements, splines, etc... which all rely on linear algebra.
I would like to stress another huge sector: the extraordinary wealth of functional analysis that has emerged after the years 1920s: normed and Banach spaces (Polish school), thorough understanding of duality culminating with the topological spaces of distributions built by Laurent Schwartz in the 1950s (French school, more or less with Bourbaki), Sobolev spaces (Russian school), etc... 
