Dual Space of V=W(+)Z: Ideas?

Nov 12, 2012

WWGD

Science Advisor

Gold Member

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?

Nov 12, 2012

Vargo

The dual of the direct sum is the direct sum of the duals.

2. 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.

3. 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.

4. 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.

5. 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.