Is the 'star' operation a vector operation for two-dimensional vectors?

[w3390]

Homework Statement

Consider a new vector operation, 'star' (*), defined for two dimensional vectors by A*B=(A1-B1,A2+B2). Is the resultant object a vector?

Homework Equations

The Attempt at a Solution

I want to say that the resultant object would be a vector but I do not know how to start to go about proving it. Any help on where I should start?

 

[Lanthanum]

It's one of those proofs that's decievingly easy.
Start by defining two general vectors;
let a=<x₁,y₁>
and b=<x₂,y₂> where x and y are real.

 

[w3390]

Okay, but I get confused on what sort of operation I should be doing since it is some sort of made up (*) operation. I already had what you just said. I didn't know where to go from there.

 

[JonF]

My proof would be something like this:

We define a vector to be an ordered pair in the form of <a,b> where a,b are reals.

Consider the binary operation on vectors A*B =(A1-B1,A2+B2), where A = <A1,A2> and B=<B1,B2>.

A1,B1,A2,B2 are all reals by the definitions of vectors. A1-B1 is a real since addition is a closed operation. A2+B2 is in the reals since addition is a closed operation. Thus (A1-B1,A2+B2) satisfies the definition of a vector.

Or do you mean an operation on a vector space? That question is a bit more fun.

 

[w3390]

What do you mean by closed operation?

 

[JonF]

What do you mean by closed operation?

Let ‘*’ be an operation defined over some space. Let a,b be ANY elements of the space. ‘*’ is said to be closed if a*b maps to a unique element in the space.

In normal human language: an operation is called closed over a set if it always produces exactly one element in the set.



But by vector do you mean an object in a vector space? If so there is a lot more to this proof.

 

[w3390]

Yes, I mean an object in vector space.

 

[JonF]

clarification: they want you to prove that * makes a vector space, or a subspace correct?

If so, this is much more fun then, but revolves around similar ideas. I would look up the definition of a vector space (should be about 8 properties) and show that the element A*B meets (or doesn’t meet) all 8 of these requirements. Remember it only has to fail one property with any elements in the set to not be a vector space.

 

[w3390]

Ok. Thanks.