Consider two linear spaces V and W. A function T from V to W is a called a linear transformation if T(f + g) = T(f) + T(g) and T(kf) = kT(f), for all elements f and g of V and for all scalars k.
The image of T is im(T) = {T(f): f in V}
The kernel of T is ker(T) = {f in V: T(f) = 0}
If V is finite dimensional, then the rank-nullity theorem holds. i.e., dim(V) = rank(T) + nullity(T) = dim(im(T)) + dim(ker T)
An isomorphism is an invertible linear transformation. We say that the linear space V is isomorphic to the linear space W if there exists an isomorphism from V to W.
Section 4.2 - 1, 5, 9, 15, 23, 37, 43