The defining two-dimensional <group representation> is <reducible representation>[reducible], because the line $\mathbb Re_1$ is an <invariant subspace>:
$$
\begin{pmatrix}a&b\\0&1\end{pmatrix}e_1=ae_1.
$$
For $b\ne0$, the complementary line $\mathbb Re_2$ is not invariant, so this representation need not split as a <direct sum> of one-dimensional representations.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 1 / d / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-302/1/d/ii/solution