The mathematical background for this statement is the linear algebra of tensor product (which is covered in my book). This is not only true of vector spaces, but also for any algebra (and the study of the results of this algebra is called ring theory, which is much better known than algebraic geometry). The importance of the special case of vector spaces is that vector spaces are the most general context in which one can define linear algebra (e.g. the study of matrix factorization), and are (in my opinion) the best place to learn to see geometric properties like dimension or the notion of a basis.
Not only I would like to suggest that the introduction be shortened, but also that the article be rewritten in a more accurate way, for instance:
to emphasize the importance of geometry and algebra in science and engineering
to emphasize the importance of geometry (in particular topology) of the spaces used in the various domains of science, engineering and mathematics
to emphasize the importance of the abstract properties of geometry (geometry =/= physics). The argumentation of the article would be more convincing if it took into account the fact that physics are based on geometry -through the linear algebra given by the tensor product-, and not in the other way around.
 The list of theorems and definitions given in section 1 are very short and sketchy. Maybe a few more specific theorems, like the generalization of the basic problem in algebraic geometry, might be specified. All theorems should be numbered.
Riddhiman and Khori have been friends since childhood. They grew up in the same house. They are best friends. Unfortunately, Khori has a hatred towards Riddhiman. Khori's sister-in-law is also in love with Riddhiman. Khori loves Aniruddh who is also friend of Riddhiman. Kunal was in love with Aniruddh. Kunal was tortured by the fact that Khori loves someone else who is in love with Aniruddh. 
The prevalence of alcohol use disorders among adults aged 18 or older, other than pregnant women, was 7.3 percent in 2016, compared with 8.3 percent in 2008, but it was similar to the estimates in 2012 to 2015 (Figure 38). The overall prevalence of substance use disorders among adults was 9.6 percent in 2016, which was higher than the estimate of 8.0 percent in 2008, but it was similar to the percentages for 2012 to 2015 (Figure 39).
In 2016, the incidence of suicidal thoughts and behaviors among youth who were current (past 12 months) users of cocaine was 1.7 per 1,000 youths, compared with 1.0 per 1,000 youths for those who were not current users (Figure 21). The incidence of suicidal thoughts and behaviors among those who were current users of marijuana was 1.4 per 1,000 youths, compared with 1.2 per 1,000 youths for those who were not current users.