all invariant subspaces
$begingroup$
Let $A$ be a linear operator such that $A begin{bmatrix}x\y\z end{bmatrix}=begin {bmatrix}x+y\y-x\0 end{bmatrix}$. Find all invariant subspaces of $A$.
I know how to find subspaces of dimensions $0$, $1$ and $3$, but how can I find all subspaces of dimension $2$ (I know that $mathrm{Im}A$ is a subspace and $mathrm{dim}; mathrm{Im} A=2$ )
linear-algebra
asked Jan 16 at 22:06
m2017mm2017m
94
$endgroup$
add a comment |
$begingroup$
Let $A$ be a linear operator such that $A begin{bmatrix}x\y\z end{bmatrix}=begin {bmatrix}x+y\y-x\0 end{bmatrix}$. Find all invariant subspaces of $A$.
I know how to find subspaces of dimensions $0$, $1$ and $3$, but how can I find all subspaces of dimension $2$ (I know that $mathrm{Im}A$ is a subspace and $mathrm{dim}; mathrm{Im} A=2$ )
linear-algebra
asked Jan 16 at 22:06
m2017mm2017m
94
$endgroup$
add a comment |
$begingroup$
Let $A$ be a linear operator such that $A begin{bmatrix}x\y\z end{bmatrix}=begin {bmatrix}x+y\y-x\0 end{bmatrix}$. Find all invariant subspaces of $A$.
I know how to find subspaces of dimensions $0$, $1$ and $3$, but how can I find all subspaces of dimension $2$ (I know that $mathrm{Im}A$ is a subspace and $mathrm{dim}; mathrm{Im} A=2$ )
linear-algebra
asked Jan 16 at 22:06
m2017mm2017m
94
$endgroup$
Let $A$ be a linear operator such that $A begin{bmatrix}x\y\z end{bmatrix}=begin {bmatrix}x+y\y-x\0 end{bmatrix}$. Find all invariant subspaces of $A$.
I know how to find subspaces of dimensions $0$, $1$ and $3$, but how can I find all subspaces of dimension $2$ (I know that $mathrm{Im}A$ is a subspace and $mathrm{dim}; mathrm{Im} A=2$ )
linear-algebra
linear-algebra
asked Jan 16 at 22:06
m2017mm2017m
94
asked Jan 16 at 22:06
m2017mm2017m
94
edited Jan 16 at 22:13
user635162
asked Jan 16 at 22:06
m2017mm2017m
94
asked Jan 16 at 22:06
m2017mm2017m
94
asked Jan 16 at 22:06
m2017mm2017m
94
94
add a comment |
add a comment |
2 Answers
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Hint: a two-dimensional subspace must intersect any other two-dimensional subspace nontrivially. Apply this to $A$ and the space generated by two basis vectors.
answered Jan 16 at 22:13
WojowuWojowu
17.9k22768
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add a comment |
$begingroup$
A two-dimensional subspace is the null space of a nonzero linear functional. The subspace is invariant if and only if the linear functional is an eigenvector of $A^*$.
answered Jan 16 at 22:16
Robert IsraelRobert Israel
322k23212465
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2 Answers
2
active
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2 Answers
2
active
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votes
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active
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votes
$begingroup$
Hint: a two-dimensional subspace must intersect any other two-dimensional subspace nontrivially. Apply this to $A$ and the space generated by two basis vectors.
answered Jan 16 at 22:13
WojowuWojowu
17.9k22768
$endgroup$
add a comment |
$begingroup$
Hint: a two-dimensional subspace must intersect any other two-dimensional subspace nontrivially. Apply this to $A$ and the space generated by two basis vectors.
answered Jan 16 at 22:13
WojowuWojowu
17.9k22768
$endgroup$
add a comment |
$begingroup$
Hint: a two-dimensional subspace must intersect any other two-dimensional subspace nontrivially. Apply this to $A$ and the space generated by two basis vectors.
answered Jan 16 at 22:13
WojowuWojowu
17.9k22768
$endgroup$
Hint: a two-dimensional subspace must intersect any other two-dimensional subspace nontrivially. Apply this to $A$ and the space generated by two basis vectors.
answered Jan 16 at 22:13
WojowuWojowu
17.9k22768
answered Jan 16 at 22:13
WojowuWojowu
17.9k22768
answered Jan 16 at 22:13
WojowuWojowu
17.9k22768
answered Jan 16 at 22:13
WojowuWojowu
17.9k22768
17.9k22768
add a comment |
add a comment |
$begingroup$
A two-dimensional subspace is the null space of a nonzero linear functional. The subspace is invariant if and only if the linear functional is an eigenvector of $A^*$.
answered Jan 16 at 22:16
Robert IsraelRobert Israel
322k23212465
$endgroup$
add a comment |
$begingroup$
A two-dimensional subspace is the null space of a nonzero linear functional. The subspace is invariant if and only if the linear functional is an eigenvector of $A^*$.
answered Jan 16 at 22:16
Robert IsraelRobert Israel
322k23212465
$endgroup$
add a comment |
$begingroup$
A two-dimensional subspace is the null space of a nonzero linear functional. The subspace is invariant if and only if the linear functional is an eigenvector of $A^*$.
answered Jan 16 at 22:16
Robert IsraelRobert Israel
322k23212465
$endgroup$
A two-dimensional subspace is the null space of a nonzero linear functional. The subspace is invariant if and only if the linear functional is an eigenvector of $A^*$.
answered Jan 16 at 22:16
Robert IsraelRobert Israel
322k23212465
answered Jan 16 at 22:16
Robert IsraelRobert Israel
322k23212465
answered Jan 16 at 22:16
Robert IsraelRobert Israel
322k23212465
answered Jan 16 at 22:16
Robert IsraelRobert Israel
322k23212465
322k23212465
add a comment |
add a comment |
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