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Could any one just tell me a sketch of the proof of the followings?

$1$. If $n>1$, does every linear transformation on $\mathbb{R}^n$ have an invariant subspaces of dimension $2$?

$2$ Is there a vector space $V$ and a linear transformation $T$ on $V$ such that it has exactly $3$ invariant subspace?

Thank you!

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up vote 3 down vote accepted
  1. Take a matrix $A \in \mathrm{GL}_n(\mathbb{R})$ that represents a linear transformation on $\mathbb{R}^n$. Think about it as a matrix over the complex numbers $A \in \mathrm{GL}_n(\mathbb{C})$. Over the complex number, $A$ has an eigenvalue $\lambda = a + ib$ (which might be complex) and an eigenvector $w \in \mathbb{C}^n$ which might be complex. Write $w = u + iv$ for $u, v \in \mathbb{R}^n$ and show that the subspace $\mathrm{span}\{u, v\}$ is an invariant subspace of dimension at most $2$ for $A$ over $\mathbb{R}^n$.
  2. The subspaces $V$ and $\{0\}$ are always invariant, so you ask whether there is a linear map which has only one non-trivial invariant subspace. This can't happen if $T$ is semi-simple, as in this case a non-trivial invariant subspace $S$ will have an invariant complement which will raise the count to at least $4$. $T$ is semi-simple if and only if the minimal polynomial of $T$ has no repeated irreducible factors. So consider the simplest non semi-simple matrix $$ \left( \begin{matrix} 0 & 1 \\ 0 & 0 \\ \end{matrix} \right) $$ whose minimal polynomial is $x^2$. The invariant subspaces are $\mathbb{R}^2$, $\mathrm{span}\{0\}$ and $\mathrm{span}\{e_1\}$. Show that there are no other invariant subspaces.
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