& Linear Transformations and Matrices
Do you believe in one vector space of dimension n? The response must be, ‘‘Yes, up to isomorphism,’’ by Theorem 3 of Section 5.2. For any given dimension n> 0, there are lots and lots and lots of diﬀerent vector spaces with that dimension, but there is also a fundamental similarity among all such spaces. No matter what objects are used to construct the space or what operations are used to combine objects, a vector space of dimension n is basically just all linear combinations of n independent objects.