2022-01-17
nick1337
Expert2022-01-19Added 777 answers
Step 1
Given complex vectors
and
their scalar (=dot) product is given by
Why? Well, you want to generalize the usual dot product on
for all v, and the vector
for all v. Most people do: the expression
should define a norm and
Having settled this, let v be a non-zero vector. Its orthogonal complement
is the set of all vectors orthogonal to v. Since U is determined by the single linear equation
it is an
which you can make into an orthonormal basis of U using Gram-Schmidt (note that the notation
is to be understood. The fact that you're working with PK\mathbb{C}\) and not with
i.e., the dot product is conjugate-linear in the second variable.
Finally, in order to solve the equation
simply take any
and put
and note that
Vasquez
Expert2022-01-19Added 669 answers
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