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Find all invariant subspaces of the linear map given by the matrix $T=\begin{bmatrix}\lambda&&1\\0&&\lambda\end{bmatrix}$ where $\lambda\in\mathbb{R}$.

The trivial invariant subspaces are $\{(0,0)\}$ and $\mathbb{R}^2$.

The only other invariant subspaces would be those spanned by the eigenvector/s of T. The only eigenvalue of T is $\lambda$ with eigenvector $(1,0)$. So then $\text{span}\{(1,0)\}$ is also an invariant subspace.

So all of the invariant subspaces are:

$\{(0,0)\}$, $\mathbb{R}^2$ and $\text{span}\{(1,0)\}$ right?

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Yes; your answer is completely correct.

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