Duality and Orthogonality: What's the Difference?

[janu203]

1. I cannot understand the difference between orthogonality and duality? Of course orthogonal vectors have dot product zero but isn't this the condition of duality as well? Swinging my head around it my cannot find the answer on the internet as well.
2.Relating to same concept is orthogonality and duality of code vectors. Does anyone have the answer?

PLZ HELP

 

[Stephen Tashi]

For one thing, orthogonality is a relation between two vectors in the same vector space. The various sorts of duality relation are usually defined between two things not in the same space. What particular definition of the relation "is the dual of" are you using?

 

[janu203]

For one thing, orthogonality is a relation between two vectors in the same vector space. The various sorts of duality relation are usually defined between two things not in the same space. What particular definition of the relation "is the dual of" are you using?

you mean to say duality is always defined between vectors of distinct spaces i.e if V1 and V2 are two vector subspaces such that V1,V2⊆V?
I want to know the duality between two vectors. What i understood is if w1 and w2 are two vectors of vector subspaces V1 and V2 respectively then w1 and w2 are dual of each other or any vector from V1 is dual of any vector from V2..

Am i right??

Orthogonality is always defined between vectors of same subspace, we cannot say w1 and w2 (above case) vectors are orthogonal.

I read somewhere Inner product is used to check if vectors are dual. Example: w1=(0 0 1 1) and w2=(0 1 0 1) then <w1,w2>=1≠0, therefore w1 amd w2 are not dual where as if a mapping function is used such that ƒ(0)=1 and ƒ(1)= -1 then <ƒ(w1),ƒ(w2)>=0 therefore they are orthogonal

What does this example tell us

 

[Stephen Tashi]

You'll have to quote an actual definition of duality to get an answer. I think you're relying on hazy memories.

A typical definition of "is the dual of" relates a vector space to the space of linear functionals on the vector space. This relation requires a mapping between vectors and linear functionals. I suppose you could say the the image of a vector under this mapping is the dual of the vector, but the usual terminology speaks of duals of spaces, not duals of individual vectors.

 

[janu203]

You'll have to quote an actual definition of duality to get an answer. I think you're relying on hazy memories.

A typical definition of "is the dual of" relates a vector space to the space of linear functionals on the vector space. This relation requires a mapping between vectors and linear functionals. I suppose you could say the the image of a vector under this mapping is the dual of the vector, but the usual terminology speaks of duals of spaces, not duals of individual vectors.

Yes! i got the duality concept , thanks

 

[Fredrik]

Yes! i got the duality concept , thanks

Do you mean that you understand the concept now after reading Stephen Tashi's post and looking up the details elsewhere, or that you already understood it? It looks like you meant the latter. In that case, you will have to think about it some more, because it's clear to us that you didn't understand the concept when you wrote the earlier posts in this thread. In particular, to say that one vector is the dual of another vector in the same space is nonsense, at least with the usual definitions. That's why Stephen asked you to post the definition you're using.

Let V be a finite-dimensional vector space over ℝ. The dual of V is the vector space V* of linear functions from V into ℝ, with addition and scalar multiplication defined in the "obvious" ways. Let $(e_i)_{i=1}^n$ be an ordered basis of V. For each i, define $e^i\in V^*$ by $e^i(e_j)=\delta^i_j$ for all j. The n-tuple $(e^i)_{i=1}^n$ is said to be the dual ordered basis of $(e_i)_{i=1}^n$.

By these standard definitions, every finite-dimensional vector space has a dual, and every ordered basis has a dual, but vectors don't have duals.

 

[janu203]

Fredrik! what my understanding of DUAL SPACE was that it is same as an orthogonal complement space where every vector of a subspace of a vector space is orthogonal to every other vector of another subspace of the same space. But replies to my thread shocked my (wrong) BELIEF so i searched out further and found out that i was wrong. Why i made this belief is due to the fact that i was reading about DUAL CODES (error correcting codes) and i misunderstood dual codes concept with the dual space . Sorry for bothering and making a mess here. Thanks