in item 2(b).\
*Case 2:* $\det (a + \lambda b) = 0$. We will show that this case cannot arise. First suppose that there is a copy of $\Omega^{l} (k)$ in $M$ for some positive integer $l$. Now pick $l$ to be the smallest such integer, then by lemma \[lemma2\] we know that $\Omega^l (k)$ is a direct summand of $M$. Since $M$ is indecomposable, this means $M$ has to be isomorphic to $\Omega^l(k)$, which is impossible since the latter is odd dimensional while $M$ was assumed to be even dimensional. So the upshot is that $M$ does not contain $\Omega^l(k)$ for any positive $l$. By lemma \[lemma1\] this is equivalent to the fact $\det (a +
\lambda b) \ne 0$ in the ring $k[\lambda]$ which contradicts our hypothesis.

Odd dimensional representations
-------------------------------

If $M$ is odd dimensional, then clearly $\dim V_1 \ne \dim V_2$. We consider the two cases.\
*Case 1:* $\dim V_1 > \dim V_2$. Then there is a non-zero vector $\omega( \lambda)$ in $ V_1 \otimes_k K[\lambda]$ such that $(a + \lambda b) (\omega (\lambda)) = 0$ which then implies, by lemma \[lemma1\], the existence of a copy of $\Omega^l(k)$ inside $M$ for some $l > 0$. 