Let $W_1, W_2$ be subspaces over $\mathbb C$ and let $V = W_1 \oplus W_2$

Assume $\langle\cdot{,}\cdot\rangle_1$ is the inner product on $W_1$ and $\langle\cdot{,}\cdot\rangle_2$ is the inner product on $W_2$.

a. find an inner product $\langle\cdot{,}\cdot\rangle$ on $V$ such that it meets the following criterias:

  • $W_2 = W_1^\perp$
  • for every $u, v \in W_k$ ($k = 1,2$) then $\langle u{,} v \rangle = \langle u {,} v \rangle_k$

b. explain if the inner product you have found is the only one possible.

my first intuition was that the standard inner product satisfies these criterias, since both subspaces are $T$ Invariant with different eigenvectors, but i still think i am missing something, mainly because of the second question... I would be happy for any explanation or clarification... thanks.

  • $\begingroup$ What do you mean by the standard inner product here? For a general vector space over $\mathbb{C}$, there is no canonical basis, so there is no notion of a standard inner product. As for your problem, try to expand $\langle u_1+u_2,v_1+v_2\rangle$ under the criterias in (a), where $u_1,v_1\in W_1$ and $u_2,v_2\in W_2$. $\endgroup$ – Batominovski Jun 20 '18 at 14:11
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    $\begingroup$ What do you mean by $\oplus$? Why not define $\langle (w_1,w_2), (v_1,v_2) \rangle = \langle w_1,v_1 \rangle + \langle w_2, v_2 \rangle$? $\endgroup$ – copper.hat Jun 20 '18 at 14:14
  • $\begingroup$ @Batominovski i was referring to the complex dot product $\endgroup$ – Limitless Jun 20 '18 at 14:14
  • $\begingroup$ @copper.hat direct sum of subspaces $\endgroup$ – Limitless Jun 20 '18 at 14:15
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    $\begingroup$ What is a standard inner product, what is $T$, are $W_1,W_2$ distinct spaces or are they subspaces of a larger ambient space? $\endgroup$ – copper.hat Jun 20 '18 at 14:22

Only specific spaces have 'standard inner product' (via their 'standard basis'), e.g. $\Bbb C^n$.

Hint: 2. is more immediate: the given conditions make a unique possibility for the value of $$\langle v_1+v_2,\ w_1+w_2\rangle$$ where $v_i\in W_i$.

For 1. prove that $\langle, \rangle$ defined this way is indeed an inner product.

  • $\begingroup$ I don't see how this can work if the $W_k$ are subsets of the same space but the subspaces are not orthogonal. The orthogonal requirement will bot be satisfied with the above then. $\endgroup$ – copper.hat Jun 20 '18 at 17:36
  • $\begingroup$ Then the inner product on the ambient space doesn't satisfy the conditions. $\endgroup$ – Berci Jun 20 '18 at 17:58
  • $\begingroup$ That was presumably the point of the question. $\endgroup$ – copper.hat Jun 20 '18 at 18:01

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