# Dimension of Free Vector Space on Set

## Theorem

Let $k$ be a division ring.

Let $X$ be a set.

Let $k^{\paren X}$ be the free vector space on $X$.

The vector space $k^{\paren X}$ 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

$\blacksquare$