Dimension of $End (V)$, $V$ infinite dimensional vector space.

Question

Let $V$ be an infinite dimensional vector space over a field $\mathbb{F}$ such that its dimension over $\mathbb{F}$ is a cardinal $k$.

Is true that the dimension of $End(V)$ as $\mathbb{F}$ vector space is again $k$?

EDIT: egreg answer below proves that if $k \geq |\mathbb{F}|$ the statement is false. But what happens in other cases? In particular what if we set $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$?

Answer 1

Let's do a specific case, with $\mathbb{F}$ the rational field or a finite field. Then, for an infinite dimensional vector space $V$, we have $$ \dim V=|V| $$ because an element of $V$ is a finite linear combination of elements of a basis and the choice of the scalars is in a countable (or finite) set.

Note now that $\lvert\operatorname{End}(V)\rvert=|V^k|$, because any endomorphism of $V$ is defined by an arbitrary function from the basis to $V$. Thus $$ \dim\operatorname{End}(V)=\lvert\operatorname{End}(V)\rvert= k^k>k=\dim V $$ The same can be said whenever $k\ge|\mathbb{F}|$.