Geometrically speaking, what "is" linear algebra?
By Sarah Richards
I have always thought of linear algebra (LA) as a system for solving linear equations. However, revisiting it after about 7-10 years, I realize that such a definition is more appropriate for "matrices" but not LA as a whole.
Although we do solve a system of linear equations to answer some questions in LA, I personally am not satisfied with that interpretation. There are a lot of (linear) transformations happening under the hood. It seems, that the definition of LA can be broadened to say that it's geometry by the way of linear transformations. That is, we "solve" for things within the system by transforming space and its containing "objects" (i.e., translate, shear, scale, rotate) vs. keeping space fixed like in a purely Cartesian setting.
Is this correct? What would be a more mathematically albeit layman friendly way to characterize this kind of geometry? It seems, everything that can be done in Euclidean Geometry can also be done in a Vector space and it's entirely possible that when in vector space if we are to find a good orthonomal basis (e.g., Eigens) we may get a much simpler representation of the problem. However, this simplification comes as a result of transforming space and seeing how the geometrical objects change along with it. Is this correct?
Would it be fair to say that (geometrically speaking) LA is a dual view of Euclidean Geometry (EG)? If so, then is there something that can be done in LA that cannot be done in EG? How could one combine the definition of LA and its relation to EG?
I could be totally wrong and hence the question to help me understand what may I be missing. I may be groping in the dark and would appreciate if I could be pointed in the right direction. None of the books (Strang, LA done right...), web/blogs, articles, history etc., that I was are able to "define" LA AFAIK.
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