So I found my way here in search of an explanation that doesn't rely on algebraic conventions, but your explanation also came to the same conclusion as every other place I've searched. My question is simply: Is there a geometric explanation for why we consider don't consider R^2 to be contained within R^3; or more generally, why don't we consider R^(n-1) to be contained within R^n? Algebraically, its obvious based on the rules of the structures in question. From a geometric standpoint it seems almost trivial to consider lower dimensional spaces as subspaces of higher-dimensional ones, and just make the definition that all vectors of finite dimensions less than n have a trail of zeros equal to n minus their dimension, especially if some subset of the basis vectors in R^n span the lower dimensional spaces. That's my rant, not directed at you, just at Math™. Great video!
Vector spaces are algebraic concepts with precise properties. they exist outside of geometry. It’s a good question, but I think it’s important to separate geometry and vector spaces - at least in terms of mathematical definitions. Your question reminds me of a story: In grad school I took a functional analysis course. I asked my professor for a geometric/visual interpretation of something he was explaining. He was confused by my question and just told me “There’s no pictures!”
@@TheLazyEngineer I understand and agree that the mathematical definition does exclude notions of geometry, and that its helpful to separate the logical structures from geometric interpretations, but I still find the notion misleading. I really appreciate the response and the anecdote!
![What is a Subspace? [Passing Linear Algebra]](http://i.ytimg.com/vi/JoeVHP_jg44/mqdefault.jpg)
So I found my way here in search of an explanation that doesn't rely on algebraic conventions, but your explanation also came to the same conclusion as every other place I've searched.
My question is simply: Is there a geometric explanation for why we consider don't consider R^2 to be contained within R^3; or more generally, why don't we consider R^(n-1) to be contained within R^n?
Algebraically, its obvious based on the rules of the structures in question. From a geometric standpoint it seems almost trivial to consider lower dimensional spaces as subspaces of higher-dimensional ones, and just make the definition that all vectors of finite dimensions less than n have a trail of zeros equal to n minus their dimension, especially if some subset of the basis vectors in R^n span the lower dimensional spaces.
That's my rant, not directed at you, just at Math™. Great video!
Vector spaces are algebraic concepts with precise properties. they exist outside of geometry. It’s a good question, but I think it’s important to separate geometry and vector spaces - at least in terms of mathematical definitions.
Your question reminds me of a story: In grad school I took a functional analysis course. I asked my professor for a geometric/visual interpretation of something he was explaining. He was confused by my question and just told me “There’s no pictures!”
@@TheLazyEngineer I understand and agree that the mathematical definition does exclude notions of geometry, and that its helpful to separate the logical structures from geometric interpretations, but I still find the notion misleading.
I really appreciate the response and the anecdote!
..Where m does not equal n. Thanks for this playlist bro!! You're helping me big time! Much appreciated!
Beautiful explanation, thank you!!
for many years I falsely assumed that it was a subset. seems like intuition and math never go hand in hand
thank you!!!
Is there any further Explanation of this topic ?
I'm looking for one as well. For my upcoming final, I am thinking of it like this: [1, 0] has z undefined. You cant say undefined = 0.
THANK YOU SO MUCH
thank you!!!