Smallest field of rationality of a subspace of a linear space

The following definitions and proposition are due to Bourbaki's Algebra Ch. II.

Let $K$ be an extension of a field $k$.

Definition 1 Let $V$ be a vector space over $K$. Let $V_0$ be a $k$-subspace of $V$. Suppose the canonical homomorphism $V_0\otimes_k K \rightarrow V$ is an isomorphism. Then $V_0$ is called a $k$-structure of $V$.

Definition 2 Let $V$ be a vector space over $K$. Let $V_0$ be a $k$-structure of $V$. Let $W$ be a $K$-subspace of $V$. If there exists a $k$-subspace $W_0$ of $V_0$ such that $W = W_0K$, then we say $W$ is rational over $k$.

Example Let $K[X_1,\dots, X_n]$ be the polynomial ring over $K$. Then $k[X_1,\dots, X_n]$ is a $k$-structure of $K[X_1,\dots, X_n]$. Let $I$ be an ideal of $K[X_1,\dots,X_n]$. Suppose $I$ is generated by polynomials of $k[X_1,\dots,X_n]$. Then $I$ is rational over $k$.

Let $V$ be a vector space over $K$. Let $V_0$ be a $k$-structure of $V$. Let $L$ be a subfield of $K$ such that $k \subset L \subset K$. Clearly $V_0L$ is a $L$-structure of $V$.

Proposition Let $V$ be a vector space over $K$. Let $V_0$ be a $k$-structure of $V$. Let $W$ be a $K$-subspace of $V$. Let $\mathfrak{F} = \{$subfield $L | k\subset L \subset K\}$. Let $\mathfrak{R} = \{L \in \mathfrak{F}| W$ is rational over $L\}$. Then $\mathfrak{R}$ has a least element.

My question Bourbaki proves the above proposition using results on rationality of a linear map. Is there a more direct proof?

EDIT Let $(v_i)_{i\in I}$ be a basis of $V_0$ over $k$. By the assumption, this is also a basis of $V$ over $K$. Since $V/W$ is generated by the set $\{v_i$ mod $W| i \in I\}$ over $K$, there exists a subset $J \subset I$ such that $(v_j$ mod $W)_{j\in J}$ is a basis of $V/W$ over $K$. For each $i \in I - J$,

$v_i \equiv \sum_{j\in J} \alpha_{ij} v_j$ mod $W$, where $\alpha_{ij} \in K$

Then we need to prove that $k(\{\alpha_{i,j}| i \in I - J, j \in J\})$ is the desired subfield.

Motivation The above proposition is important in arithmetic theory of algebraic varieties as the above example shows.

@QiL xbvn in the above link just stated that, in our notation, $k(\{\alpha_{i,j}| i \in I - J, j \in J\})$ is the desired subfield. I would like to know its proof. – Makoto Kato Nov 19 '12 at 23:08