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.
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.