Basis

Before, we emphasized a keyword, ‘create’, and say an arbitrary vector can be created by two non-overlapping vectors in 2D space. Now, I want to replace this keyword with ‘represented’, maybe a more formal one. With this new keyword, we move our attention from “resulting vector” to “the two original linearly independent vectors”. We call them a set of basis in the sense that they are the backbone of the space constituted by all possible vectors.

I have a metaphor that may not particularly fit. The basis is like the various departments under the student union of a university, such as the study department, the culture and sports department, the life department, and so on. The students in these departments are by no means all students in this university, but they can represent all students very well. If a student is dissatisfied with the university, it is best not to quarrel with the dean alone. The smartest way is to negotiate with the institute through the student union. Keep this in mind! Creator: Alejandro A. Alvarez | Credit: Alejandro A. Alvarez / Staff Photographer

Let’s summarize. In a 2D space, a pair of non-overlapped vectors can be a basis. So the angle between the two vectors is very essential. Then how do we deal with angles in linear algebra? Another question is can we find a better basis in the sense that the coefficients of linear combination can be easily obtained? These questions lead to the next important operation, the inner product.

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