(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let V={a cosx + b sinx | a,b [tex]\in[/tex] R}

(a) Show that V is a subspace of the R-vector space of all maps from R to R.

(b) Show that V is isomorphic to R^2, under the map

f: V[tex]\rightarrow[/tex]R^2

a cosx + b sinx [tex]\rightleftharpoons[/tex] [ a over b ] (this is supposed to be a matrix with a above and b below, couldn't find it in the Latex reference)

3. The attempt at a solution

I have done part (a) okay so it's just part (b) I need a hand with. So I know V is isomorphic to R^2 if the map f is linear and the dimV = dim R^2.

I think I sort of showed that the dimensions are equal by taking a basis of {0 , cosx + sinx}. Is that basis okay? And am I right in thinking that because that has two elements dimV=2 and obviously dimR^2= 2 yeah?

So assuming that all that's ok so far I'm kind of stuck showing that it's linear. The properties of a linear map are f(u+v)=f(u)+f(v) and f(av)= af(v) where u,v[tex]\in[/tex]V and a[tex]\in[/tex] R. I'm trying to show those properties are true at the minute but having some difficulty. Am I on the right track at least?

thanks in advance!

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# Linear map is isomorphic

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