Thread: Determine wheter a subset is a subspace.

Hi, I've done some exercises to determine wheter a subset is a subspace, but I don't know how to do the ones with functions, like the following:

Determine wheter the following subset is a subspace of the vector space of all the functions f(x), x ∈ R.

G={acosx+bsinx: a, b ∈ R}

So, I have to check that G satisfies the additive identity, the closed under addition and closed under escalar properties.

I think the first condition is satisfied, right? How do I check if the other two are satisfied?

Hey Cesc1.

Hint: What is the definition of addition and scalar multiplication in the vector space? (Also what is the exact definition of a vector in the space)?

Remember, what are the elements of G? Are they real numbers? Are they functions?

How can you add two functions together? How can you multiply a function by a scalar?

Originally Posted by chiro

Hey Cesc1.

Hint: What is the definition of addition and scalar multiplication in the vector space? (Also what is the exact definition of a vector in the space)?

A vector space V over a field F is a set V equipped with an operation called (vector) addition, which takes vectors u and v and produces another vector u + v . There is also an operation called scalar multiplication, which takes an element a in F and a vector u in V and produces a vector au in V.

So, if I take vectors u and v from G, such that u= a1cosx + b1sinx, and v= a2cosx + b2sinx, their sum u+v= (a1cosx+b1sinx)+(a2cosx+b2sinx) can be expressed as (a1+a2)cosx + (b1+b2)sinx, so the closed under addition property is satisfied.

Also, if I have u= acosx + bsinx, and some number c from R, then cu= c(acosx + bsinx)= (ac)cosx + bc(sinx), which is also in G. So, I think that G is indeed a subspace. Is that right?

Yes, that is correct!