Are These Sets Vector Spaces?

[madking153]

hi,i got 2 question about vector spaces :

1. Do the set of all n-tuples of real numbers of the form (x, x1 ,x2...xn) with the standard operation on R^2 are vector spaces?

2.Do the set of all positive real numbers with operations

x+y =x*y and kx=x^2 are vector space?

 

[matt grime]

1 makes no sense. What do "the standard operations of R^2", which are undefined, have to do with the n-tuples of real numbers? R^n is a vector space.

2, just try and verify the axioms, or figure out where they may go wrong.

 

[madking153]

those question is from a book ...

 

[HallsofIvy]

And does the book use phrases like "those question"?

Once again: your first question makes no sense. It would make sense if you asked "Does the set of all n-tuples of real numbers of the form (x1 ,x2...xn) with the standard operation on R^n form a vector space?". In that case the anser is obviously "yes". It makes no sense to talk about " real numbers of the form (x1 ,x2...xn) with the standard operation on R^2" because you can't apply the operations on R^2 to R^n.

The second question, "2.Is the set of all positive real numbers with operations
x+y =x*y and kx=x^2 a vector space?" is reasonable. Matt Grimes' point was that it is just a matter of checking the axioms for (or definition of) a vector space.

 

[madking153]

so you are saying that :
the set of all positive real numbers with operations
x+y =x*y and kx=x^2 are vector space?

 

[HallsofIvy]

No, we are saying you should check to see if the "axioms" for a vector space are satisfied yourself. In particular, is the "distributive law", k(x+y)= kx+ ky, satisfied?

 

[madking153]

( a 1 ) --- is this 2x2 matrix a vector space ? sorry for asking this coz this is abstract
1 b me

 

[matt grime]

Why is it abstract? What are the operations, for instance, what is k*M for some scalar k? Is that in the set? Check the rules, there really is nothing complicated or hidden in this:

to check if S is a vector space over R, say, check the rules: is ks in S when s is in S and k is in R? ie if s satisfies the rules to be in S, does ks? Similarly what about s+t when s and t are in S? is there a zero vector? Just three simple rules to check.

 

[Data]

there are a few more rules to check too (like the afformentioned distributive property).