Determining if a set is a subspace

[mlarson9000]

Homework Statement

Determine whether or not the set of all functions f such that f(1)+f(-1)=f(5) is a subspace of the vector space F of all functions mapping R into R.

Homework Equations

The Attempt at a Solution

I think that
(f(1)+f(-1))+(g(1)+g(-1))=(f+g)(1)+(f+g)(-1)=(f+g)(5)
shows it is closed under vector addition, but I'm not sure. I'm also not sure what to do about checking scalar multiplication. I don't think that r(f(1)+f(-1)=rf(5) does anything at all for me.

 

[slider142]

Your demonstration of closure under vector addition is fine. With respect to scalar multiplication, you are being asked to show that rf defined as (rf)(x) = r*f(x) is also an element of this vector space whenever f is.

 

[mlarson9000]

So (rf)(1)+(rf)(-1)=r*(f(1)+f(-1))= rf(5) which is not equal to f(5), so it's not closed under scalar multiplication?

 

[slider142]

So (rf)(1)+(rf)(-1)=r*(f(1)+f(-1))= rf(5) which is not equal to f(5), so it's not closed under scalar multiplication?

Why are we looking at f(5)? The defining axiom is that h(1) + h(-1) = h(5) for each h in the vector space.

 

[mlarson9000]

so do I only need to show (rf)(1)+(rf)(-1)=r*(f(1)+f(-1))?

 

[slider142]

so do I only need to show (rf)(1)+(rf)(-1)=r*(f(1)+f(-1))?

You also need to show that this is equivalent to (rf)(5), which may seem trivial, but necessary.