Basis of vector space vs coordinate system, changing them and non-linear transformations

This question has been asked before and I've got an idea of the difference but i'm still not completely certain at points and want to be meticulous on the definitions to help me connect the dots, here's what I currently understand

• $$\mathbb{R}^n$$ with component-wise addition and the scalar being an element from $$\mathbb{R}$$ that scales all components of an element in the set defines a vector space

• A basis is a set of vectors that allow us to completely describe the vector space.

• We can say that $$\mathbb{R}^n$$ has an "inherent" basis which uniquely assigns each vector a coordinate. (By inherent I mean the vector itself can be considered it's own coordinate)

• By choosing a different basis (redefining what vector is associated with $$(1,0,\dots,0)^T$$ etc ) we can uniquely assign vectors a different coordinate which is equivalent to a reshuffling which elements of $$R^n$$ represent which vector.

• This change of basis can be interpreted as a linear transformation $$x_i = \Phi(q_1,q_2,\dots,q_n)$$ where $$x_i$$ represents the $$i^{th}$$ component of the vectors coordinate in the new basis and $$(q_1,q_2,\dots,q_n)^T$$ is the coordinate in the old basis. An alternative way to write this would be $$\textbf{x}=\textbf{Aq}$$ where $$\textbf{A}$$ is an $$nxn$$ matrix.

• Linear transformations preserve algebraic operations so we can still add and scale like we did before. This however can not be said if $$\Phi$$ was a non-linear function, however regardless of what coordinates we use to represent vectors the notion of being able to add and scale remains unchanged.

• A coordinate system refers to assigning each point on a n-dimensional manifold an n-tuple coordinate that discerns the points position - some points may have multiple coordinates associated with it.

• An n-dimensional affine space is isomorphic to $$\mathbb{R}^n$$. Which allows us to assign each point on the affine space a coordinate that coincides with the coordinate of the vector. As such we can use a basis to induce a coordinate system for affine space.

• Specifically, by picking some arbitrary origin in affine space $$O$$ and a basis set $$\mathcal{B}$$ we can define a coordinate system $$(O,\mathcal{B}$$).

• Typically the coordinate of the vector and the coordinate of the point coincide. By doing this and using $$(O,\mathcal{B})$$ we preserve algebraic properties of the vector space so that the coordinates of points and vectors can be added component wise.

• If we assigned coordinates to a vector in a different way (say a non-linear transformation of coordinates from a coordinate system that uses basis vectors) and assigned a point with the same coordinate as its vector the notion of "adding a vector to a point" remains the same but the process in which we do it changes.