Simple inner product question on CMOplex SPace

[trap101]

Let the inner product on P₂(C) be defined as:

∫ (from -1 to 1) p(t)(conjugate) q(t) dt.

using the gram schmidt process and the standard basis {1, x, x²} find an orthonormal basis.


So my only issue really is interpreting this integral. I just wanted to test if the vectors in the basis were not orthogonal for my own assurance, but now I'm not sure if that is the case. So taking the inner product of < x , 1>:

< x, 1> = x²/2 ? is that right? I mean there are no complex values in the standard basis so I don't see how the conjugate is changing anything.

 

[Dick]

Let the inner product on P₂(C) be defined as:

∫ (from -1 to 1) p(t)(conjugate) q(t) dt.

using the gram schmidt process and the standard basis {1, x, x²} find an orthonormal basis.


So my only issue really is interpreting this integral. I just wanted to test if the vectors in the basis were not orthogonal for my own assurance, but now I'm not sure if that is the case. So taking the inner product of < x , 1>:

< x, 1> = x²/2 ? is that right? I mean there are no complex values in the standard basis so I don't see how the conjugate is changing anything.

Yes, the conjugate isn't changing anything. NO, <x,1> is NOT x^2/2. It's a definite integral.

 

[trap101]

Yes, the conjugate isn't changing anything. NO, <x,1> is NOT x^2/2. It's a definite integral.

Right, but now this is a past exam question I'm working on, and if that is the case. The set of standard basis vectors would each be orthogonal to each other. Which makes sense, then all I would have to do is "normalize" them to make them orthonormal. But that is only 3-4 lines at most. I don't know something seems odd about that.

 

[Dick]

Right, but now this is a past exam question I'm working on, and if that is the case. The set of standard basis vectors would each be orthogonal to each other. Which makes sense, then all I would have to do is "normalize" them to make them orthonormal. But that is only 3-4 lines at most. I don't know something seems odd about that.

<1,x^2> isn't 0. What is it?

 

[trap101]

<1,x^2> isn't 0. What is it?

True, thank god. I'm having another small calculation question on another inner product deinfed as:

<p(x),q(x)> = P(0)(conjugate) q(0) + P(i)(conjugate) q(i)

now the question is asking to find an orthonormal basis for P₁(C)

going through the gram schmidt again, I'm getting tangled up in an inner product that isn't giving me what I think it should.

So using the standard basis of P₁: {1,x} I let "1" be my initial vector. So to find the next vector call it v₂:

v2 = x - < x, 1>/<1,1> so I'm having issues with < x , 1 >.

Doesn't < x, 1> = (-i)/2 ? thus giving me a basis vector of (x + i/2)...this didn't turn out being orthogonal

 

[Dick]

True, thank god. I'm having another small calculation question on another inner product deinfed as:

<p(x),q(x)> = P(0)(conjugate) q(0) + P(i)(conjugate) q(i)

now the question is asking to find an orthonormal basis for P₁(C)

going through the gram schmidt again, I'm getting tangled up in an inner product that isn't giving me what I think it should.

So using the standard basis of P₁: {1,x} I let "1" be my initial vector. So to find the next vector call it v₂:

v2 = x - < x, 1>/<1,1> so I'm having issues with < x , 1 >.

Doesn't < x, 1> = (-i)/2 ? thus giving me a basis vector of (x + i/2)...this didn't turn out being orthogonal

Be careful with the order of the arguments to the inner product. <x,1>=i. <1,x>=(-i). Try it carefully. And maybe I have that wrong. Which function is the conjugate on, p or q?

 

[trap101]

Be careful with the order of the arguments to the inner product. <x,1>=i. <1,x>=(-i). Try it carefully. And maybe I have that wrong. Which function is the conjugate on, p or q?

Great. That means the way I had accepted of finding ortogonal bases over the complex is flawed. I had worked the general form as being <v2, u1>/<u1, u1> because I had an issue with this before but over the regularly defined complex inner product and that brought about the right solution, the textbook had it defined as <u1,v2>/<u1,u1> but then doing it that way I didn't get the right solution.

Looks like I'm going to have to try both and see which satisfies it in the proper way.