I read this sentence in Linear Algebra and its Applications: "Linear transformations preserve the operations of vector addition and scalar multiplication," but I don't know preserve means in this context. Alas, googling it and looking in the book's index found nothing relevant.
So, what does it mean?
Best Answer
It means it doesn't matter whether you do the operation first and the transformation second, or the transformation first and the operation second. You get the same result either way.
So for example for vector addition, if $f$ preserves vector addition then for any vectors $u, v$ in the domain of $f$ we have $$f(u + v) = f(u) + f(v).$$ On the left we did addition first, on the right we did the transformation first, and the equation says you get the same thing either way. For scalar multiplication the equation is $$f(cv) = cf(v)$$ for all vectors $v$ and scalars $c$.