Dimension of Free Vector Space on Set

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Theorem

Let $k$ be a division ring.

Let $X$ be a set.

Let $k^{(X)}$ be the free vector space on $X$.


The vector space $k^{(X)}$ has dimension the cardinality of $X$.


Proof

Follows from:

Canonical Basis of Free Module on Set is Basis
Cardinality of Canonical Basis of Free Module on Set

$\blacksquare$