Inner Product Space of two orthogonal Vectors is 0 , Is this defined as it is ?

[vish_maths]

This may be a very silly question, but still apologies, I read in Sheldon Axler, that the inner product of two orthogonal vectors is DEFINED to be 0.
Let u,v belong to C^n. I am unable to find a direction of proof which proves that for an nth dimension vector space, if u perp. to v, then <u,v> = 0
Is it really just defined ? Or it can be proved to be 0 ?

 

[micromass]

This may be a very silly question, but still apologies, I read in Sheldon Axler, that the inner product of two orthogonal vectors is DEFINED to be 0.

This is not correct. Rather, we define two vectors to be orthogonal if their inner product is 0.
So given $u,v\in \mathbb{C}^n$, we say that u and v are orthogonal iff <u,v>=0. This is a definition.

 

[HallsofIvy]

It can be shown, in R^{2 or R3, where we have a geometric definition of "orthogonal", that two vectors are orthogonal if and only if their dot product is 0. For higher dimension Euclidean spaces or more general vector spaces, it is simplest to take "inner product is 0" as the definition of "orthogonal".}

 

[vish_maths]

Thanks a lot :). There has been just one more question which has been lingering in my mind.

If V is a complex inner product space and T is an operator on V such that <Tv,v> = 0 for all v belongs to V. Then T =0.
Though, it's proof is somewhat convincing , it has left me confused about
a) the existence of orthogonality in n dimensional vectors belong to C^n
b) if the answer above is yes, then why can't Tv be orthogonal to v. ( even if it's not visual, I mean in the mathematical sense)

 

[blue_raver22]

I can confirm that the inner product of two orthogonal vectors is indeed defined as 0. This definition is based on the properties of inner product spaces, where the inner product is a function that takes two vectors and produces a scalar value. In this case, the definition states that the inner product of two orthogonal vectors will always be 0, regardless of the dimension of the vector space.

However, this definition can also be proved using mathematical proofs and properties of inner product spaces. For example, in a complex vector space, the inner product is defined as <u,v> = u* * v, where u* represents the complex conjugate of u. Therefore, if u and v are orthogonal, their inner product would be <u,v> = u* * v = 0, since the complex conjugate of a number multiplied by its orthogonal complement is always 0.

In summary, the inner product of two orthogonal vectors is both defined to be 0 and can be proved to be 0 using mathematical proofs and properties of inner product spaces.