**1. The problem statement, all variables and given/known data**
Not really a problem per se; more of an issue with some aspects of linear transformations. We've learned that a linear combination of linear transformations is defined as follows:

[tex](c_1T_1+c_2T_2)(\vec{x})=c_1T_1(\vec{x})+c_2T_2(\vec{x})\,\,\,\,\,\, \vec{x}\varepsilon \mathbb{R}^n\,\,\,c\varepsilon \mathbb{R}[/tex]

And if T

_{1}, T

_{2} are linear, then so is c

_{1}T

_{1} + c

_{2}T

_{2} and

[tex][c_1T_1+c_2T_2]=c_1[T_1]+c_2[T_2][/tex]

Where [T] is the standard matrix of T and c is some constant.

The problem is, is that I don't know why T

_{1} and T

_{2} being linear implies that c

_{1}T

_{1} + c

_{2}T

_{2} is linear other than by noting that multiplying the transformations by c and adding them together is a linear combination, but that doesn't seem to be a very good proof. The second part there would seem to follow from this though just based on the fact that transforming some vector x is the same as multiplying it by the standard matrix of T.

Also, linear transformations don't make much sense in general. I was told that they're basically the same things as functions, but then all of a sudden we're adding them together and multiplying them left and right which never happened with f(x). Does that matter? Can I just think of those

*T*s as

*f*s? If so, I think that will help a lot because, as it is, the

*T* seems very strange and foreign to me.

I would appreciate any tips on such matters