Vector subspace, what dimension could it be?
$begingroup$
I'm new here and hope that I can help also others with questions, but now I have one, I was three weeks sick and wasn't often in the university and now I'm just wondering how to do my task.
Let $U$, $V$ and $W$ be $2$-dimensional subspaces of $mathbb{R}^6$ with
$$U + V = U ⊕ V,quad U + W = U ⊕ W,quadtext{and}quad V + W = V ⊕ W$$
What dimension could $U+V+W$ have?
Here we should give a example but how I do that? Can give someone me some tips?
I have also another task which is really similar.
Let $U$, $V$ and $W$ be $1$-dimensional subspaces of $mathbb{R}^3$ with
$$U + V = U ⊕ V,quad U + W = U ⊕ W,quadtext{and}quad V + W = V ⊕ W$$
but $U + V + W ≠ U ⊕ V ⊕ W$
Also here I should give a example and explain why.
So please can give me someone a tip?
Thanks from now :)
linear-algebra
edited Jan 13 at 14:03
egreg
181k1485202
asked Jan 13 at 13:38
HanniHanni
31
$endgroup$
add a comment |
$begingroup$
I'm new here and hope that I can help also others with questions, but now I have one, I was three weeks sick and wasn't often in the university and now I'm just wondering how to do my task.
Let $U$, $V$ and $W$ be $2$-dimensional subspaces of $mathbb{R}^6$ with
$$U + V = U ⊕ V,quad U + W = U ⊕ W,quadtext{and}quad V + W = V ⊕ W$$
What dimension could $U+V+W$ have?
Here we should give a example but how I do that? Can give someone me some tips?
I have also another task which is really similar.
Let $U$, $V$ and $W$ be $1$-dimensional subspaces of $mathbb{R}^3$ with
$$U + V = U ⊕ V,quad U + W = U ⊕ W,quadtext{and}quad V + W = V ⊕ W$$
but $U + V + W ≠ U ⊕ V ⊕ W$
Also here I should give a example and explain why.
So please can give me someone a tip?
Thanks from now :)
linear-algebra
edited Jan 13 at 14:03
egreg
181k1485202
asked Jan 13 at 13:38
HanniHanni
31
$endgroup$
add a comment |
$begingroup$
I'm new here and hope that I can help also others with questions, but now I have one, I was three weeks sick and wasn't often in the university and now I'm just wondering how to do my task.
Let $U$, $V$ and $W$ be $2$-dimensional subspaces of $mathbb{R}^6$ with
$$U + V = U ⊕ V,quad U + W = U ⊕ W,quadtext{and}quad V + W = V ⊕ W$$
What dimension could $U+V+W$ have?
Here we should give a example but how I do that? Can give someone me some tips?
I have also another task which is really similar.
Let $U$, $V$ and $W$ be $1$-dimensional subspaces of $mathbb{R}^3$ with
$$U + V = U ⊕ V,quad U + W = U ⊕ W,quadtext{and}quad V + W = V ⊕ W$$
but $U + V + W ≠ U ⊕ V ⊕ W$
Also here I should give a example and explain why.
So please can give me someone a tip?
Thanks from now :)
linear-algebra
edited Jan 13 at 14:03
egreg
181k1485202
asked Jan 13 at 13:38
HanniHanni
31
$endgroup$
I'm new here and hope that I can help also others with questions, but now I have one, I was three weeks sick and wasn't often in the university and now I'm just wondering how to do my task.
Let $U$, $V$ and $W$ be $2$-dimensional subspaces of $mathbb{R}^6$ with
$$U + V = U ⊕ V,quad U + W = U ⊕ W,quadtext{and}quad V + W = V ⊕ W$$
What dimension could $U+V+W$ have?
Here we should give a example but how I do that? Can give someone me some tips?
I have also another task which is really similar.
Let $U$, $V$ and $W$ be $1$-dimensional subspaces of $mathbb{R}^3$ with
$$U + V = U ⊕ V,quad U + W = U ⊕ W,quadtext{and}quad V + W = V ⊕ W$$
but $U + V + W ≠ U ⊕ V ⊕ W$
Also here I should give a example and explain why.
So please can give me someone a tip?
Thanks from now :)
linear-algebra
linear-algebra
edited Jan 13 at 14:03
egreg
181k1485202
asked Jan 13 at 13:38
HanniHanni
31
edited Jan 13 at 14:03
egreg
181k1485202
asked Jan 13 at 13:38
HanniHanni
31
edited Jan 13 at 14:03
egreg
181k1485202
edited Jan 13 at 14:03
egreg
181k1485202
edited Jan 13 at 14:03
egreg
181k1485202
181k1485202
asked Jan 13 at 13:38
HanniHanni
31
asked Jan 13 at 13:38
HanniHanni
31
asked Jan 13 at 13:38
HanniHanni
31
31
add a comment |
add a comment |
2 Answers
2
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oldest
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$begingroup$
For the former: we know that $dim(U + V) = dim(Uoplus V) = 4$, so $U + V + W$ has dimension at least 4. Indeed, it can also be exactly four: with $e_1$ through $e_6$ the standard basis vectors for $mathbb{R}^6$, take $U = langle e_1, e_2rangle$, $V = langle e_3,e_4rangle$, $W = langle e_1+e_3,e_2+e_4rangle$, and note that $e_1,e_2,e_3,e_4$ is a basis for $U + V + W = U + V = U + W = V + W$. On the other side, $U + V + W subseteq mathbb{R}^6$, so $dim (U + V + W) leq 6$. It should be easy to construct examples where $dim U$ is each of $5$ and $6$.
The example I gave also satisfies your second question.
answered Jan 13 at 14:08
user3482749user3482749
4,206919
$endgroup$
add a comment |
$begingroup$
Hint 1
$U + V + W$ has Dimension at most $6$.
Hint 2
$U + V + W$ has Dimension at least $4$, as it contains...
Hint 3
Start with $U, V$, and try and construct an example with $W subseteq U + V$.
Hint 4
In $W$ you may try and put suitable elements of the form $u + v$, with $0 ne u in U$, $0 ne v in V$.
answered Jan 13 at 14:11
Andreas CarantiAndreas Caranti
56.5k34395
$endgroup$
add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
For the former: we know that $dim(U + V) = dim(Uoplus V) = 4$, so $U + V + W$ has dimension at least 4. Indeed, it can also be exactly four: with $e_1$ through $e_6$ the standard basis vectors for $mathbb{R}^6$, take $U = langle e_1, e_2rangle$, $V = langle e_3,e_4rangle$, $W = langle e_1+e_3,e_2+e_4rangle$, and note that $e_1,e_2,e_3,e_4$ is a basis for $U + V + W = U + V = U + W = V + W$. On the other side, $U + V + W subseteq mathbb{R}^6$, so $dim (U + V + W) leq 6$. It should be easy to construct examples where $dim U$ is each of $5$ and $6$.
The example I gave also satisfies your second question.
answered Jan 13 at 14:08
user3482749user3482749
4,206919
$endgroup$
add a comment |
$begingroup$
For the former: we know that $dim(U + V) = dim(Uoplus V) = 4$, so $U + V + W$ has dimension at least 4. Indeed, it can also be exactly four: with $e_1$ through $e_6$ the standard basis vectors for $mathbb{R}^6$, take $U = langle e_1, e_2rangle$, $V = langle e_3,e_4rangle$, $W = langle e_1+e_3,e_2+e_4rangle$, and note that $e_1,e_2,e_3,e_4$ is a basis for $U + V + W = U + V = U + W = V + W$. On the other side, $U + V + W subseteq mathbb{R}^6$, so $dim (U + V + W) leq 6$. It should be easy to construct examples where $dim U$ is each of $5$ and $6$.
The example I gave also satisfies your second question.
answered Jan 13 at 14:08
user3482749user3482749
4,206919
$endgroup$
add a comment |
$begingroup$
For the former: we know that $dim(U + V) = dim(Uoplus V) = 4$, so $U + V + W$ has dimension at least 4. Indeed, it can also be exactly four: with $e_1$ through $e_6$ the standard basis vectors for $mathbb{R}^6$, take $U = langle e_1, e_2rangle$, $V = langle e_3,e_4rangle$, $W = langle e_1+e_3,e_2+e_4rangle$, and note that $e_1,e_2,e_3,e_4$ is a basis for $U + V + W = U + V = U + W = V + W$. On the other side, $U + V + W subseteq mathbb{R}^6$, so $dim (U + V + W) leq 6$. It should be easy to construct examples where $dim U$ is each of $5$ and $6$.
The example I gave also satisfies your second question.
answered Jan 13 at 14:08
user3482749user3482749
4,206919
$endgroup$
For the former: we know that $dim(U + V) = dim(Uoplus V) = 4$, so $U + V + W$ has dimension at least 4. Indeed, it can also be exactly four: with $e_1$ through $e_6$ the standard basis vectors for $mathbb{R}^6$, take $U = langle e_1, e_2rangle$, $V = langle e_3,e_4rangle$, $W = langle e_1+e_3,e_2+e_4rangle$, and note that $e_1,e_2,e_3,e_4$ is a basis for $U + V + W = U + V = U + W = V + W$. On the other side, $U + V + W subseteq mathbb{R}^6$, so $dim (U + V + W) leq 6$. It should be easy to construct examples where $dim U$ is each of $5$ and $6$.
The example I gave also satisfies your second question.
answered Jan 13 at 14:08
user3482749user3482749
4,206919
answered Jan 13 at 14:08
user3482749user3482749
4,206919
answered Jan 13 at 14:08
user3482749user3482749
4,206919
answered Jan 13 at 14:08
user3482749user3482749
4,206919
4,206919
add a comment |
add a comment |
$begingroup$
Hint 1
$U + V + W$ has Dimension at most $6$.
Hint 2
$U + V + W$ has Dimension at least $4$, as it contains...
Hint 3
Start with $U, V$, and try and construct an example with $W subseteq U + V$.
Hint 4
In $W$ you may try and put suitable elements of the form $u + v$, with $0 ne u in U$, $0 ne v in V$.
answered Jan 13 at 14:11
Andreas CarantiAndreas Caranti
56.5k34395
$endgroup$
add a comment |
$begingroup$
Hint 1
$U + V + W$ has Dimension at most $6$.
Hint 2
$U + V + W$ has Dimension at least $4$, as it contains...
Hint 3
Start with $U, V$, and try and construct an example with $W subseteq U + V$.
Hint 4
In $W$ you may try and put suitable elements of the form $u + v$, with $0 ne u in U$, $0 ne v in V$.
answered Jan 13 at 14:11
Andreas CarantiAndreas Caranti
56.5k34395
$endgroup$
add a comment |
$begingroup$
Hint 1
$U + V + W$ has Dimension at most $6$.
Hint 2
$U + V + W$ has Dimension at least $4$, as it contains...
Hint 3
Start with $U, V$, and try and construct an example with $W subseteq U + V$.
Hint 4
In $W$ you may try and put suitable elements of the form $u + v$, with $0 ne u in U$, $0 ne v in V$.
answered Jan 13 at 14:11
Andreas CarantiAndreas Caranti
56.5k34395
$endgroup$
Hint 1
$U + V + W$ has Dimension at most $6$.
Hint 2
$U + V + W$ has Dimension at least $4$, as it contains...
Hint 3
Start with $U, V$, and try and construct an example with $W subseteq U + V$.
Hint 4
In $W$ you may try and put suitable elements of the form $u + v$, with $0 ne u in U$, $0 ne v in V$.
answered Jan 13 at 14:11
Andreas CarantiAndreas Caranti
56.5k34395
answered Jan 13 at 14:11
Andreas CarantiAndreas Caranti
56.5k34395
answered Jan 13 at 14:11
Andreas CarantiAndreas Caranti
56.5k34395
answered Jan 13 at 14:11
Andreas CarantiAndreas Caranti
56.5k34395
56.5k34395
add a comment |
add a comment |
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