I see this a lot with math concepts as they begin to get more abstract: strange visualizations to try to build intuition. I think this is ultimately a dead-end approach which misleads rather than enlightens.
To me, the proper way of continuing to develop intuition is to abandon visualization entirely and start thinking about the math in a linguistic mode. Thus, continuous functions (perhaps on the closed interval [0,1] for example) are vectors precisely because this space of functions meet the criteria for a vector space:
* (+) vector addition where adding two continuous functions on a domain yields another continuous function on that domain
* (.) scalar multiplication where multiplying a continuous function by a real number yields another continuous function with the same domain
* (0) the existence of the zero vector which is simply the function that maps its entire domain of [0,1] to 0 (and we can easily verify that this function is continuous)
We can further verify the other properties of this vector space which are:
* associativity of vector addition
* commutativity of vector addition
* identity element for vector addition (just the zero vector)
* additive inverse elements (just multiply f by -1 to get -f)
* compatibility of scalar multiplication with field multiplication (i.e a(bf) = (ab)f, where a and b are real numbers and f is a function)
* identity element for scalar multiplication (just the number 1)
* distributivity of scalar multiplication over vector addition (so a(f + g) = af + ag)
* distributivity of scalar multiplication over scalar addition (so (a + b)f = af + bf)
So in other words, instead of trying to visualize an infinite-dimensional space, we’re just doing high school algebra with which we should already be familiar. We’re just manipulating symbols on paper and seeing how far the rules take us. This approach can take us much further when we continue on to the ideas of normed vector spaces (abstracting the idea of length), sequences of vectors (a sequence of functions), and Banach spaces (giving us convergence and the existence of limits of sequences of functions).
To me, the proper way of continuing to develop intuition is to abandon visualization entirely and start thinking about the math in a linguistic mode. Thus, continuous functions (perhaps on the closed interval [0,1] for example) are vectors precisely because this space of functions meet the criteria for a vector space:
* (+) vector addition where adding two continuous functions on a domain yields another continuous function on that domain
* (.) scalar multiplication where multiplying a continuous function by a real number yields another continuous function with the same domain
* (0) the existence of the zero vector which is simply the function that maps its entire domain of [0,1] to 0 (and we can easily verify that this function is continuous)
We can further verify the other properties of this vector space which are:
* associativity of vector addition
* commutativity of vector addition
* identity element for vector addition (just the zero vector)
* additive inverse elements (just multiply f by -1 to get -f)
* compatibility of scalar multiplication with field multiplication (i.e a(bf) = (ab)f, where a and b are real numbers and f is a function)
* identity element for scalar multiplication (just the number 1)
* distributivity of scalar multiplication over vector addition (so a(f + g) = af + ag)
* distributivity of scalar multiplication over scalar addition (so (a + b)f = af + bf)
So in other words, instead of trying to visualize an infinite-dimensional space, we’re just doing high school algebra with which we should already be familiar. We’re just manipulating symbols on paper and seeing how far the rules take us. This approach can take us much further when we continue on to the ideas of normed vector spaces (abstracting the idea of length), sequences of vectors (a sequence of functions), and Banach spaces (giving us convergence and the existence of limits of sequences of functions).