Proof of sums of linear transformations

[veritaserum20]

Given linear transformations S: Rn --> Rm and T: Rn --> Rm, show the following:
a) S+T is a linear transformation
b) cS is a linear transformation

I know that since both S and T are linear transformations on their own, they satisfy the properties for being a linear transformation, which is that for some transformation T, T(x+y)=T(x) + T(y), and T(cx)=cT(x). So I tried doing the same sort of procedure for the sum of the transformations, so that (S+T)(x)=S(x) + T(x) and S(cx)=cS(x). This just doesn't seem like a very intricate way of proving the sums of linear transformations is a linear transformation and that a scalar multiplied by a linear transformation is a linear transformation.

Any help is greatly appreciated!

 

[radou]

This just doesn't seem like a very intricate way of proving the sums of linear transformations is a linear transformation and that a scalar multiplied by a linear transformation is a linear transformation.

Why not? You defined (S+T)(x) = S(x) + T(x). Now you only need to check if S+T is a linear transformation, i.e. that for all x, y and [tex]\alpha[/tex], [tex]\beta[/tex] , (S+T)([tex]\alpha[/tex]x+[tex]\beta[/tex]y)=[tex]\alpha[/tex](S+T)(x) + [tex]\beta[/tex](S+T)(y) holds.

 

[HallsofIvy]

There is no requirement that proofs be "intricate"!