How Can the Dual of V in Terms of Z and W Be Expressed?

In summary, the dual space of V=W(+)Z is the set of all linear functionals on the vector space V=W(+)Z. It is based on the concept of duality and is intimately related to the original vector space. Some real-world applications include physics, engineering, and economics. Common misconceptions include equating it to the transpose of a matrix and assuming it is always finite-dimensional.
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WWGD
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Hi, All:

Let V be a finite-dimensional space, which can be decomposed as:

V=Z(+)W . How can we express the dual of V in terms of the duals of

Z, W?

I think this has to see with tensor products, but I'm kind of rusty here.

Any ideas, please?
 
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  • #2
The dual of the direct sum is the direct sum of the duals.
 

FAQ: How Can the Dual of V in Terms of Z and W Be Expressed?

What is the dual space of V=W(+)Z?

The dual space of V=W(+)Z refers to the set of all linear functionals on the vector space V=W(+)Z. In other words, it is the space of all linear maps from V=W(+)Z to its underlying field, typically denoted by V=W(+)Z* or V=W(+)Z'.

What are the main ideas behind the concept of dual space?

The concept of dual space is based on the idea of duality, which states that every vector space has a corresponding "dual" space of linear functionals. This allows for a deeper understanding of the underlying structure of a vector space and enables the use of powerful mathematical tools such as dual bases, dual maps, and dual transformations.

How is the dual space of V=W(+)Z related to the original vector space?

The dual space of V=W(+)Z is intimately related to the original vector space in that it contains all the information about the linear functionals on V=W(+)Z. Furthermore, the dual space can be seen as a "mirror image" of the original space, with vectors in the dual space representing functionals on the original space.

What are some real-world applications of the dual space of V=W(+)Z?

The dual space of V=W(+)Z has various applications in fields such as physics, engineering, and economics. For example, in physics, the dual space is used to describe the state of a physical system and its observable properties. In economics, the dual space is used to model the preferences of individuals and firms in decision-making processes.

What are some common misconceptions about the dual space of V=W(+)Z?

One common misconception is that the dual space is the same as the transpose of a matrix. While the transpose of a matrix can be used to represent a linear transformation, it is not the same as the dual space. Another misconception is that the dual space is always finite-dimensional, when in reality it can be infinite-dimensional in many cases.

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