Proof- Vector fields form vector space

In summary, to prove that the set of all planar vector fields forms a vector space, you must start by defining what a planar vector field is and then compare it to the definition of a vector space. This can be done by looking at the different vector operations and seeing which operations on planar vector fields correspond to them. Additionally, you can use the fact that if V is a vector space and S is a set, then the set of functions from S to V is also a vector space.
  • #1
deanachuz
1
0
How can I prove that the set of all planar vector fields forms a vector space? Thanks for any input!
 
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  • #2
Welcome to PF;

You need to start with the definition of a planar vector field - how would you tell if a particular field were a member of the set?

Then you need to compare this with the definition of a vector space ... which operations on planar vector fields would correspond to the different vector operations.

Presumably you've already seen how to do this with some examples that don't seem, at first glance, to be vectors ... like polynomials?
 
  • #3
if V is a vector space and S is a set, then the set of functions S-->V is a vector space.
 

What is a vector field?

A vector field is a mathematical concept that associates a vector (a direction and magnitude) to each point in a given space.

What is a vector space?

A vector space is a set of vectors that satisfy specific properties, such as closure under addition and scalar multiplication.

How do we know that vector fields form a vector space?

We can prove that vector fields form a vector space by showing that they satisfy the axioms of a vector space, such as closure under addition and scalar multiplication, and the existence of a zero vector and additive inverses.

What are some real-life examples of vector fields?

Some real-life examples of vector fields include wind speed and direction, electric and magnetic fields, and fluid flow velocity fields.

Why is it important to understand that vector fields form a vector space?

Understanding that vector fields form a vector space allows us to apply various mathematical concepts and techniques to analyze and manipulate them, making it a powerful tool in many scientific fields such as physics, engineering, and computer graphics.

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