How does the dimension of two subspaces relate to their intersection?Possible dimensions of the intersection of two subspacesProve dimension of sum of two subspacesdimension of intersection of subspacesdimension of intersection of a family of subspacesDimension of sum of subspacesIntersection of subspaces and dimensionsCourant–Fischer Theorem Proof [Meyer]. Intersection of two subspaces with same dimension.Dimension of intersection of three subspacesDimension of intersection of 2 distinct subspaces of codimension 1.Intersection of subspaces in $mathbbR^4$
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How does the dimension of two subspaces relate to their intersection?
Possible dimensions of the intersection of two subspacesProve dimension of sum of two subspacesdimension of intersection of subspacesdimension of intersection of a family of subspacesDimension of sum of subspacesIntersection of subspaces and dimensionsCourant–Fischer Theorem Proof [Meyer]. Intersection of two subspaces with same dimension.Dimension of intersection of three subspacesDimension of intersection of 2 distinct subspaces of codimension 1.Intersection of subspaces in $mathbbR^4$
$begingroup$
Suppose $U$ and $V$ are subspaces of $mathbb R^8$ such that $dim, U=3$, $dim, V=5$, and $U cup V=mathbb R^8$. Prove that $Uoplus V=mathbb R^8$.
My question is:
How can I, or if I should, use the dimension of $U$ and $V$ to show that their intersection is empty?
linear-algebra vector-spaces
edited Mar 13 at 19:18
6005
37k751127
asked Mar 13 at 19:11
noobiskonoobisko
715
$endgroup$
add a comment |
$begingroup$
Suppose $U$ and $V$ are subspaces of $mathbb R^8$ such that $dim, U=3$, $dim, V=5$, and $U cup V=mathbb R^8$. Prove that $Uoplus V=mathbb R^8$.
My question is:
How can I, or if I should, use the dimension of $U$ and $V$ to show that their intersection is empty?
linear-algebra vector-spaces
edited Mar 13 at 19:18
6005
37k751127
asked Mar 13 at 19:11
noobiskonoobisko
715
$endgroup$
1
$begingroup$
Intersection is not empty. It Just have the zero element.
$endgroup$
– dmtri
Mar 13 at 19:14
add a comment |
$begingroup$
Suppose $U$ and $V$ are subspaces of $mathbb R^8$ such that $dim, U=3$, $dim, V=5$, and $U cup V=mathbb R^8$. Prove that $Uoplus V=mathbb R^8$.
My question is:
How can I, or if I should, use the dimension of $U$ and $V$ to show that their intersection is empty?
linear-algebra vector-spaces
edited Mar 13 at 19:18
6005
37k751127
asked Mar 13 at 19:11
noobiskonoobisko
715
$endgroup$
Suppose $U$ and $V$ are subspaces of $mathbb R^8$ such that $dim, U=3$, $dim, V=5$, and $U cup V=mathbb R^8$. Prove that $Uoplus V=mathbb R^8$.
My question is:
How can I, or if I should, use the dimension of $U$ and $V$ to show that their intersection is empty?
linear-algebra vector-spaces
linear-algebra vector-spaces
edited Mar 13 at 19:18
6005
37k751127
asked Mar 13 at 19:11
noobiskonoobisko
715
edited Mar 13 at 19:18
6005
37k751127
asked Mar 13 at 19:11
noobiskonoobisko
715
edited Mar 13 at 19:18
6005
37k751127
edited Mar 13 at 19:18
6005
37k751127
edited Mar 13 at 19:18
6005
37k751127
37k751127
asked Mar 13 at 19:11
noobiskonoobisko
715
asked Mar 13 at 19:11
noobiskonoobisko
715
asked Mar 13 at 19:11
noobiskonoobisko
715
715
1
$begingroup$
Intersection is not empty. It Just have the zero element.
$endgroup$
– dmtri
Mar 13 at 19:14
add a comment |
1
$begingroup$
Intersection is not empty. It Just have the zero element.
$endgroup$
– dmtri
Mar 13 at 19:14
1
1
$begingroup$
Intersection is not empty. It Just have the zero element.
$endgroup$
– dmtri
Mar 13 at 19:14
$begingroup$
Intersection is not empty. It Just have the zero element.
$endgroup$
– dmtri
Mar 13 at 19:14
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This formula should help:
$$
dim(U + V) = dim(U) + dim(V) - dim(U cap V).
$$
What can you conclude about $dim(U cap V)$? What does that imply about $U oplus V$?
answered Mar 13 at 19:17
60056005
37k751127
$endgroup$
2
$begingroup$
P.S. Just a random aside, but the analogous statement for three vector spaces ($dim(U + V + W) = cdots$) is not true; see this famous MathOverflow post.
$endgroup$
– 6005
Mar 13 at 19:27
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
This formula should help:
$$
dim(U + V) = dim(U) + dim(V) - dim(U cap V).
$$
What can you conclude about $dim(U cap V)$? What does that imply about $U oplus V$?
answered Mar 13 at 19:17
60056005
37k751127
$endgroup$
2
$begingroup$
P.S. Just a random aside, but the analogous statement for three vector spaces ($dim(U + V + W) = cdots$) is not true; see this famous MathOverflow post.
$endgroup$
– 6005
Mar 13 at 19:27
add a comment |
$begingroup$
This formula should help:
$$
dim(U + V) = dim(U) + dim(V) - dim(U cap V).
$$
What can you conclude about $dim(U cap V)$? What does that imply about $U oplus V$?
answered Mar 13 at 19:17
60056005
37k751127
$endgroup$
2
$begingroup$
P.S. Just a random aside, but the analogous statement for three vector spaces ($dim(U + V + W) = cdots$) is not true; see this famous MathOverflow post.
$endgroup$
– 6005
Mar 13 at 19:27
add a comment |
$begingroup$
This formula should help:
$$
dim(U + V) = dim(U) + dim(V) - dim(U cap V).
$$
What can you conclude about $dim(U cap V)$? What does that imply about $U oplus V$?
answered Mar 13 at 19:17
60056005
37k751127
$endgroup$
This formula should help:
$$
dim(U + V) = dim(U) + dim(V) - dim(U cap V).
$$
What can you conclude about $dim(U cap V)$? What does that imply about $U oplus V$?
answered Mar 13 at 19:17
60056005
37k751127
answered Mar 13 at 19:17
60056005
37k751127
answered Mar 13 at 19:17
60056005
37k751127
answered Mar 13 at 19:17
60056005
37k751127
37k751127
2
$begingroup$
P.S. Just a random aside, but the analogous statement for three vector spaces ($dim(U + V + W) = cdots$) is not true; see this famous MathOverflow post.
$endgroup$
– 6005
Mar 13 at 19:27
add a comment |
2
$begingroup$
P.S. Just a random aside, but the analogous statement for three vector spaces ($dim(U + V + W) = cdots$) is not true; see this famous MathOverflow post.
$endgroup$
– 6005
Mar 13 at 19:27
2
2
$begingroup$
P.S. Just a random aside, but the analogous statement for three vector spaces ($dim(U + V + W) = cdots$) is not true; see this famous MathOverflow post.
$endgroup$
– 6005
Mar 13 at 19:27
$begingroup$
P.S. Just a random aside, but the analogous statement for three vector spaces ($dim(U + V + W) = cdots$) is not true; see this famous MathOverflow post.
$endgroup$
– 6005
Mar 13 at 19:27
add a comment |
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$begingroup$
Intersection is not empty. It Just have the zero element.
$endgroup$
– dmtri
Mar 13 at 19:14