I'm in a linear algebra class and am having a hard time wrapping my head around what subspaces of a vector space are useful for (among many other things!). My understanding of a vector space is that, simplistically, it defines a coordinate plane that you can plot points on and figure out some useful things about the relationship between vectors/points.
I think what I'm curious about is more application of some of these ideas. Such as, is a subspace useful for a reason other than you don't have to look at the entire space something exists in (I guess one way I've been thinking about it is if you want to make a map of a city, you don't necessarily need to make a map of the state it's in) or am I even wrong about that much? Also, even though I feel like I should know this at this point, is if the subspace is linearly independent, is it still a subspace? If it is, what exactly does that describe and/or why is that still useful? If it's not, is it still useful for something?
I think the most difficult part of this for me is I'm having a hard time being able to visualize what exactly we're talking about and I have a hard time thinking that abstractly. I know one or two examples of this might be too specific and doesn't generalize the concept enough, but I think if I have some example to relate back to when applying the idea to new things it might be helpful.
I really appreciate any thoughts or responses about this! :)