Linear Independence and Dependence in C^d: Questions from Abbas Edalat's Notes

[G.F.Again]

Questions from Abbas Edalat's notes
http://www.doc.ic.ac.uk/~ae/teaching.html#quantum"

It's about vector space, linear independence and linear dependence of vectors or something else.

Thanks!

Generalize the following notions and properties given for the vector space C^2 in the notes to C^d.
1) Define the norm \omega of a vector and the inner product of two vectors \omega(1) and \omega(2) in C^d. What is the dual of a vector \omega in C^d and what can it be identified with?

2) Define linear independence and linear dependence of vectors in C^d.

3) What is the least integer n such that any set of n vectors in C^d will be linearly dependent?

4) What is a basis of C^d? How many linearly independent vectors it takes to get a basis for C^d?

5) Define the notion of an orthonormal basis for C^d. What would be the standard basis of C^d?

 

[HallsofIvy]

What have you tried? What are the definitions of those things in C^3?

 

[G.F.Again]

I'm sorry, but I really want to say is as the file bellow. Thanks again.

 

[HallsofIvy]

1. Many people will not download a "Word" document from someone they do not know.

2. I did go ahead and look at it, against my better judgement, and it adds nothing- it's just a statement of the problems. You have not responded to any of my questions: What have you tried? What are the definitions of those things in C^2?

 

[G.F.Again]

Thanks anyway, HallsofIvy. I think I can do that now.
What I define the C^2, C^d, R^2, and D^d have the meaning that C^2 is a 2-dimension complex Hilbert space, and C^d a d-dimension complex Hilbert space; similarly,R^2 is a 2-dimension real space, and R^d a d-dimension real number space.And the norm \omega is the norm of omega.