I had studied it under a similar approach. Of course, after taking Abstract Algebra course, and seeing Polynomials over Abstract Fields, which form a vector space, while computing the degree of finite field extensions[1], I realized how much powerful Linear Algebra was.
Also during a Computational Complexity course, when we were studying de-randomization techniques, computing the Spectrum of an expander graph[2] and looking into the eigenvalues and eigenvectors of the graphs' adjacency matrix to understand topological properties from graphs, I realized how powerful and general Linear Algebra can be.
2d and 3d vectors over the Field of Real Numbers are easy to see, and a good way to get started. But many beautiful and powerful things come from this theory.
Also during a Computational Complexity course, when we were studying de-randomization techniques, computing the Spectrum of an expander graph[2] and looking into the eigenvalues and eigenvectors of the graphs' adjacency matrix to understand topological properties from graphs, I realized how powerful and general Linear Algebra can be.
2d and 3d vectors over the Field of Real Numbers are easy to see, and a good way to get started. But many beautiful and powerful things come from this theory.
[1] http://en.wikipedia.org/wiki/Degree_of_a_field_extension
[2] http://en.wikipedia.org/wiki/Expander_graph