Let $φ: \mathbb{R^2} \to \mathbb{R^2}$ be a linear transformation and $A = \begin{bmatrix}1&-2\\2&2 \end{bmatrix}$ be its representation with respect to the standard basis. Find all of the $φ-$invariant subspaces.

{$0$} and $\mathbb{R^2}$ are trivial invariant subspaces of dimensions $0$ and $2$.

For Dimension $1$, I've shown that there's no invariant subspace of dimension $1$. So, the only invariant subspaces are {$0$} and $\mathbb{R^2}$. Are these really the only ones? Do I need to show there are no other ones?

  • 1
    $\begingroup$ Any subspace of $\Bbb R^2$ must have dimension $0,1$ or $2$ where $0$ is for $\{0\}$ and $2$ is for $\Bbb R^2$. If you have shown that there doesn't exist any 1-dimensional invariant subspace, then yes, you're done. $\endgroup$ – Prasun Biswas Jan 12 '18 at 11:49
  • $\begingroup$ Is $\mathbb{R^2}$ the only invariant subspace of dimension $2$ ? $\endgroup$ – Alex Matt Jan 12 '18 at 12:06
  • $\begingroup$ Not just invariant, $\Bbb R^2$ is the only subspace of dimension $2$ since any subspace of $\Bbb R^2$ with two linearly independent generators must span $\Bbb R^2$ $\endgroup$ – Prasun Biswas Jan 12 '18 at 12:10

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