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- Proving $(A\\to B)\\to\\perp$ $\\Rightarrow$ $A\\land(B\\to\\perp . . .
@DanChristensen I'm afraid I don't get it There're many propositions that are equivelant but cost many steps to prove, for example A → B A → B ⇒ ⇒ (A →⊥) ∧ B (A →⊥) ∧ B (7 blocks, task 5 5) Would you give a detailed answer, either by word or by using incredible pm? So I can understand you and close the question? Thanks
- linear algebra - Show that $ (U + W)^ {\perp} = U^ {\perp}\cap W . . .
If U U and W W are subspaces of a finite dimensional inner product space V V, show that (a) If U ⊆ W U ⊆ W, then W⊥ ⊆ U⊥ W ⊥ ⊆ U ⊥ (b) (U + W)⊥ =U⊥ ∩W⊥ (U + W) ⊥ = U ⊥ ∩ W ⊥ (c) U⊥ +W⊥ ⊂ (U ∩ W)⊥ U ⊥ + W ⊥ ⊂ (U ∩ W) ⊥ MY ATTEMPT (a) Let {u1,u2, …,um} {u 1, u 2,, u m} be a basis for U U Since U U is a subspace of W W, we can extend such
- Finding a subspace $W \\subset \\mathbb{R}^4$ such that $W \\subset S . . .
You'll need to complete a few actions and gain 15 reputation points before being able to upvote Upvoting indicates when questions and answers are useful What's reputation and how do I get it? Instead, you can save this post to reference later
- Show a linear operator - Mathematics Stack Exchange
Let T T be a linear operator between two normed spaces I'm trying to show that an operator T T has closed range if and only if im(T) = (ker(T∗))⊥ im (T) = (ker (T ∗)) ⊥ Is there a way to do it without the Hahn-Banach theorem? Thanks for any help
- $M$ and $N$ are subspaces of a Hilbert space. If $M\\subset N$, show . . .
M M and N N are subspaces of a Hilbert space If M ⊂ N M ⊂ N, show that N⊥ ⊂M⊥ N ⊥ ⊂ M ⊥ Show also that (M⊥)⊥ = M (M ⊥) ⊥ = M I know that the orthogonal complement of X X is the set X⊥ = {x ∈ H: x⊥X} X ⊥ = {x ∈ H: x ⊥ X} where X X is any subset of a Hilbert space H H I'm not sure how to proceed Any hints or solutions are greatly appreciated
- Symplectic gradient is just giving me the gradient
This will give the skew gradient in coordinates, and in the case that a compatible metric structure exists, one may turn this into "perp of the gradient" by the identity ♯ω = J ∘♯g ♯ ω = J ∘ ♯ g I believe the presence of Hodge star in the original notes is a quirk of J J coinciding with −⋆g ⋆ g on 1-forms in R2 R 2
- linear algebra - How to show that $ (W^\bot)^\bot=W$ (in a finite . . .
I need to prove that if V V is a finite dimensional vector space over a field K with a non-degenerate inner-product and W ⊂ V W ⊂ V is a subspace of V, then:
- The range of $T^*$ is the orthogonal complement of $\\ker(T)$
How can I prove that, if V V is a finite-dimensional vector space with inner product and T T a linear operator in V V, then the range of T∗ T ∗ is the orthogonal complement of the null space of T T? I know what I must do (for a v v in the range of T∗ T ∗, I have to show that v ⊥ w v ⊥ w for every w w in ker(T) ker (T) and then do the opposite), but I don't know how to show that
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