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

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In summary, the conversation is about generalizing notions and properties for the vector space C^2 to C^d. It covers topics such as defining the norm and inner product of vectors in C^d, the dual of a vector in C^d, linear independence and linear dependence of vectors, the basis of C^d, the least integer n for linear dependence, and the notion of an orthonormal basis for C^d. The definitions of C^2, C^d, R^2, and R^d are also mentioned.
  • #1
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?
 
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  • #2
What have you tried? What are the definitions of those things in C^3?
 
  • #3
I'm sorry, but I really want to say is as the file bellow. Thanks again.
 

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  • #4
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?
 
  • #5
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.
 

1. What is the definition of linear independence and dependence?

Linear independence and dependence refer to the relationship between vectors in a vector space. A set of vectors is considered linearly independent if no vector in the set can be written as a linear combination of the other vectors. On the other hand, a set of vectors is considered linearly dependent if at least one vector in the set can be expressed as a linear combination of the other vectors.

2. How can we determine if a set of vectors is linearly independent or dependent?

One way to determine the linear independence or dependence of a set of vectors is by using the determinant method. This involves creating a matrix using the vectors and calculating the determinant. If the determinant is equal to 0, then the vectors are linearly dependent. If the determinant is not equal to 0, then the vectors are linearly independent.

3. Can a set of more than d vectors be linearly independent in C^d?

No, in C^d, a set of vectors can be linearly independent only if the number of vectors in the set is equal to or less than d. This is because a vector in C^d has d components, so it can be expressed as a linear combination of at most d other vectors.

4. What happens if one vector in a set of linearly dependent vectors is removed?

If one vector is removed from a set of linearly dependent vectors, the remaining vectors will still be linearly dependent. This is because the removed vector can be expressed as a linear combination of the remaining vectors, and therefore, it does not add any new information to the set.

5. Are all bases in C^d considered to be linearly independent?

Yes, all bases in C^d are considered to be linearly independent. This is because a basis is a set of vectors that span the entire vector space, and any set of vectors that can span a vector space must be linearly independent. However, not all linearly independent sets of vectors are considered to be bases, as they may not span the entire vector space.

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