Is V Isomorphic to R^2 Under the Given Mapping?

[Mitch_C]

Homework Statement

Let V={a cosx + b sinx | a,b $$\in$$ 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$$\rightarrow$$R^2
a cosx + b sinx $$\rightleftharpoons$$ [ a over b ] (this is supposed to be a matrix with a above and b below, couldn't find it in the Latex reference)

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$$\in$$V and a$$\in$$ 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!

 

[Office_Shredder]

How do you construct sinx from the set {0, cosx + sinx}?

Also any set containing zero is not linearly independent since 1*0=0 so there is a non-trivial linear combination which gives zero.

 

[HallsofIvy]

Homework Statement

Let V={a cosx + b sinx | a,b $$\in$$ 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$$\rightarrow$$R^2
a cosx + b sinx $$\rightleftharpoons$$ [ a over b ] (this is supposed to be a matrix with a above and b below, couldn't find it in the Latex reference)

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?

No, it is NOT okay. A basis never includes the 0 vector. Try {cos x, sin x} instead.

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$$\in$$V and a$$\in$$ 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!

"\begin{bmatrix} a \\ b\end{bmatrix}" gives
$$\begin{bmatrix} a \\ b\end{bmatrix}$$

If u= a cos(x)+ b sin(x) and v= c cos(x)+ d sin(x), what is u+ v?

$$f(u)= \begin{bmatrix}a \\ b\end{bmatrix}$$
and
$$f(v)= \begin{bmatrix}c \\ d \end{bmatrix}$$

what is f(u+ v)?

 

[Mitch_C]

Thanks for that. Looking back on it the basis I picked is obviously not a basis I just wasn't thinking. Spelling out for me what f(u) equals really helped. When I came back to it I solved it in about 10mins!

Thanks again! :)