How Can Scalars Represent Any Linear Transformation in R^3?

[loli12]

Let T:R^3 -> R be linear. Show that there exist scalars a, b, and c such that T(x, y , z) = ax + by + cz for all (x, y, z) in R^3. State and prove an analogous result for T: F^n -> F^m.

I know that we just have to multiply by a matrix then we can get the desired transformation. But how would I go around to show that such scalars a, b and c exists?

 

[Galileo]

Use the fact that T is linear. This means that T(v+w)=T(v)+T(w).
So if you know a basis for R^3 (there's an obvious one) and you know how T acts on this basis, you know how T acts on every vector in R^3.