The orthogonal complement of the empty set is indeed V. However, the theorem that (U^\bot)^\bot=U has as a hypothesis that U is a subspace of V. The empty set is not a subspace of V and thus is not an exception to this result.
Thanks for your brilliant videos, professor. It seems all of the proofs here makes the assumption that the vector space is finite dimensional. Would you give some nonexamples in which the theorems are not true for infinite dimensional vector space?
See, for example, Exercise 14 in Section 6.C of the book for an example of a result about orthogonal complements that does not hold in the infinite-dimensional case.
Excuse me, Sheldon Axler, sir, how do we initially come up with the idea to let the coefficient in the linear combination be the scalar production of v and each component of the basis {e1,e2,....em}?
Your question is answered in the proof of 6.30 in the book. Note that Chapter 6 of the book is freely available on the book's website (linear.axler.net/).
In the book, in the proof of 6.51, and given 6.52, wouldn't it be easier to prove that dim((Uperp)perp) = dim(U)? Indeed dim((Uperp)perp) + dim(Uperp) = dim(V) and dim(U) + dim(Uperp) = dim(V).
very intuitive and neat explanation, professor.
indeed, your lectures seem much more intuitive and clear than the Kenneth Hoffman linear algebra book!
Thank you so much for the videos!
What is the complement of the empty set? If it's V, then we have an exception to (U^\bot)^\bot=U because (\emptyset^\bot)^\bot=\{0\}
The orthogonal complement of the empty set is indeed V. However, the theorem that (U^\bot)^\bot=U has as a hypothesis that U is a subspace of V. The empty set is not a subspace of V and thus is not an exception to this result.
Thanks for your brilliant videos, professor. It seems all of the proofs here makes the assumption that the vector space is finite dimensional. Would you give some nonexamples in which the theorems are not true for infinite dimensional vector space?
See, for example, Exercise 14 in Section 6.C of the book for an example of a result about orthogonal complements that does not hold in the infinite-dimensional case.
Excuse me, Sheldon Axler, sir, how do we initially come up with the idea to let the coefficient in the linear combination be the scalar production of v and each component of the basis {e1,e2,....em}?
Your question is answered in the proof of 6.30 in the book. Note that Chapter 6 of the book is freely available on the book's website (linear.axler.net/).
In the book, in the proof of 6.51, and given 6.52, wouldn't it be easier to prove that dim((Uperp)perp) = dim(U)? Indeed dim((Uperp)perp) + dim(Uperp) = dim(V) and dim(U) + dim(Uperp) = dim(V).
6.51 doesn’t state v is finite dimensional so u cannot use 6.50.
@@mkkkk1643neither holds in infinite dimensional
@spiderjerusalem4009 just simply read the proof