This came up today when people showed that there is no linear transformation $\mathbb{R}^4\to \mathbb{R}^3$.
However, we know that these sets have the same cardinality. I was under the impression that if two sets have the same cardinality then there exists a bijection between them. Is this true? Or is it just that any two sets which have a bijection between them have the same cardinality.
Edit: the question I linked to is asking specifically about a linear transformation. My question still holds for arbitrary maps.
Answer
"Same cardinality" is defined as meaning there is a bijection.
In your vector space example, you were requiring the bijection to be linear. If there is a linear bijection, the dimension is the same. There is a bijection between $\mathbb R^4$ and $\mathbb R^3$, but no such bijection is linear, or even continuous. (Space-filling curves, which are continuous functions from a space of lower dimension to a space of higher dimension, are not bijections since they are in no instance one-to-one.) If there is a bijection, then the cardinality is the same. And conversely.
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