= Solution
The <four-subspace quiver> has four one-dimensional sources and a two-dimensional sink. Its <Tits form of a quiver> at this dimension vector is $q=4\cdot1^2+2^2-4(1\cdot2)=0$.
An <endomorphism> comprises source scalars $a_1,\ldots,a_4$ and a sink matrix $T$. The first two columns $e_1,e_2$ force $T=\operatorname{diag}(a_1,a_2)$. The third column $e_1+e_2$ then forces $a_1=a_2=a_3$, making $T$ scalar. Since the fourth column $(\lambda,1)^T$ is always nonzero, its scalar is also the same, for every $\lambda$.
Thus $\operatorname{End}_Q(V)=k$, so the representation is a <brick module>. The <Ringel form> gives
$$
\boxed{\dim\operatorname{Ext}^1_Q(V,V)=\dim\operatorname{End}_Q(V)-q=1}.
$$
This conclusion also covers $\lambda=0,1$, where the fourth line repeats one of the earlier lines; the first three lines already force scalar endomorphisms.
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