The discussion revolves around verifying calculations related to the dot product in a complex vector space and finding a specific function \( h \) that satisfies certain conditions. The original poster presents their findings on the norms and inner products of two vectors \( f \) and \( g \), and raises questions about the implications of the span of these vectors.
Several participants have provided feedback on the calculations, indicating areas where the original poster's work may be incorrect or needs clarification. The conversation is ongoing, with multiple interpretations of the problem being explored and no explicit consensus reached on the final values or the function \( h \).
There are discussions about the definition of the inner product and its properties, particularly in relation to complex numbers. Participants are also clarifying the notation used for spans and the implications of the results obtained in the context of the problem.
dirk_mec1 said:The instructor has put a pdf file on the internet (which I didn't see), apparently the inner product is defined as above:
http://img221.imageshack.us/img221/2899/58958904pd7.png
So then the answer is ||g|| = \sqrt{ \frac{1}{3} }
But for <f,g> the inner product can have two values depending which function you use for the overbar, right?
I talked to the instructor and the expression above is incorrect.gabbagabbahey said:Hmmm...I'm not too familiar with the notation here. What exactly do the []'s mean?
Here is how I would write the span:
\text{span} \{ f,g \} =\{ \alpha_n \left( \frac{-1}{2} \right) ^n+ \beta_n \left( \frac{i}{2} \right) ^n|\alpha_n,\beta_n \in \mathbb{C} \quad \forall \; n \in \mathbb{N} \}
HallsofIvy said:How many definitions of "inner product" are there?
The only one I know is
"An inner product on a vector space V is a mapping from VxV to C (or R is the vector space is over R) such that
1) <au+ bv,w>= a<u,v>+ b<v,w>
2) <u, v>= complex conjugate of <v, u> (or just <v,u> if the vector space is over R)
3) <u, u> \ge 0
4) <u, u>= 0 if and only if u= 0."
If by "defining" the inner product, you mean defining such a function on a specific vector space, if it does not satisfy those, it is NOT an inner product.
HallsofIvy said:Yes, you can. The innerproduct, over the complex numbers, is NOT symmetric:
<f, g>= \overline{<g, f>}