= Solution
Every <vector space> has a <vector space basis>[basis] and is classified up to <linear isomorphism> by its <dimension of a vector space>[dimension]. Over a <finite field>, a vector is a finite <linear combination> of <vector space basis>[basis] <Elements>. If the <vector space basis>[basis] has infinite <cardinality> $\lambda$, the <set> of finite such combinations has <cardinality> $\lambda$: there are at least $\lambda$ <basis vectors> and at most a <countable union> of finite powers of $\lambda$, each of size $\lambda$. Thus the underlying <cardinality> of an infinite space equals its <dimension of a vector space>[dimension].
Any two <first-order model>[models] of <cardinality> $\kappa\ge\aleph_0$ therefore have <vector space basis>[bases] of <cardinality> $\kappa$. A <bijection> between their <vector space basis>[bases] extends uniquely to an $L$-isomorphism. A space with that <vector space basis>[basis] size exists, giving the required existence clause as well. Hence \b[$T$ is categorical in every infinite cardinal].
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