TheWrathOfMath
Junior Member
- Joined
- Mar 31, 2022
- Messages
- 162
If the subspaces V,U,W ⊂ R^6, and dimV=dimW=5, dimU=3, then U⋂V⋂W={0}
I know that the answer is no, and I can take U ⊆ V=W, but I managed to find an example for V,U,W ⊆ R^6 instead of V,U,W ⊂ R^6.
W=V={(1 0 0 0 1 0) (0 1 0 0 0 0) (0 0 1 0 0 0) (0 0 0 1 0 0) (0 0 0 0 0 1) (1 1 1 1 1 1)}
dimU=dimV=5
U= {(1 1 0 0 0 0) (0 0 1 1 0 0) (0 0 0 0 1 1)}
dimU=3
U⋂V⋂W =/= {0}
The problem is that, as stated above, V,U,W ⊆ R^6 instead of V,U,W ⊂ R^6.
How do I fix this?
I know that the answer is no, and I can take U ⊆ V=W, but I managed to find an example for V,U,W ⊆ R^6 instead of V,U,W ⊂ R^6.
W=V={(1 0 0 0 1 0) (0 1 0 0 0 0) (0 0 1 0 0 0) (0 0 0 1 0 0) (0 0 0 0 0 1) (1 1 1 1 1 1)}
dimU=dimV=5
U= {(1 1 0 0 0 0) (0 0 1 1 0 0) (0 0 0 0 1 1)}
dimU=3
U⋂V⋂W =/= {0}
The problem is that, as stated above, V,U,W ⊆ R^6 instead of V,U,W ⊂ R^6.
How do I fix this?