Question

what do we infer from the study of vector spaces in linear algebra and where does it applies in this real world?

Answer 1

???????? ok What?

Answer 2

There are TONS of reasons to study linear algebra. Vector spaces provide deep insight into elegant methods of approximating functions (i.e. Fourier series) and solving differential equations (i.e. eigenvalues & eigenvectors). Additionally, vector spaces allow us to know a great deal about solution spaces by finding an appropriate basis and then representing all other solutions as linear combinations of those basis vectors.

I'm not sure how much of that makes sense, there was an awful lot of vocabulary, but I would recommend checking out this link to see eigenvalues & eigenvectors in action:
http://ocw.mit.edu/ans7870/18/18.03/s06/tools/LinPhasePorMatrix.html

Try the following:
1 Click and drag in the yellow/green/blue area
2 Click the "Eigenvalues" button
3 Click in the large graph area to draw lines of attraction/repulsion
4 Click "Clear" to clear the large graph
5 Drag the sliders.