Dimension of Subspaces of Euclidean Space

We prove that in a subspace of Euclidean space, R^n, the number of elements of any basis is an invariant. This invariant cardinality will be defined as the dimension of the subspace. We show that with this definition of dimension, the dimension of R^n is as we expect, n. We provide context for this definition and explain the difficulties moving beyond finite dimensions.
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