Interpretation of linear transformations Am I right in saying that all linear transformations can be thought of as a combination of scaling and rotation? How does one prove/disprove this statement?
 A: Scaling (by nonzero scalars) and rotations have something in common: they are all bijections of the plane. Of course, there are many linear transformations from a vector space $V$ to $V$ that aren't one-to-one or surjective. As folks have mentioned already, linear projections and reflections are obviously omitted types of linear transformation from the set you described (and there are a lot more missing, too.)
What you are talking about is a special subgroup of the set of all linear transformations. One special illustration of what you're describing is the complex numbers: they describe rotations and dilations of the complex plane. However, complex conjugation is another linear transformation beyond those. You could also zero out the $y$ coordinate and leave the $x$ coordinate alone and flatten the plane onto the $x$-axis, and that's a linear transformation too.
A: "Linear transformation" is a much more general term. T is linear iff T(x+py)=T(x)+pT(y) for all vectors x,y and every scalar p. Examples: (1) T(x)=0 for all x.  (2) T(x,y)=x for all x,y .  (3)$ T(x,y)=(x,-y)$ for all x,y. Example 3, when x,y are real numbers, turns the outline of a left glove into the outline of a right one, which you cannot do by scaling and rotation. 
