Definition. A map f from a vector space V to a vector space W is linear if it obeys the two rules
f(x_{1} + x_{2}) = f(x_{1}) + f(x_{2}) for all x_{1}, x_{2} in V.
f(c x) = c f(x) for all scalars c and all x in V.
Five Examples
If M is an m \times n matrix and L_{M}(x) = M x for x in R^{n}, then L_{M} is a linear map from R^{n} to R^{m}.
If F is the vector space of all differentiable functions on the interval a < x < b, then the map D defined by D(f) = {df}/{dt} is a linear map from F to the set of functions on the interval a < x < b.
If S is the vector space of all solutions of the differential equation f'' = f, then the map phi from S to R^{2} defined by phi(f) = (f(0), f'(0)) is a linear map.
If S is the vector space of all solutions of the differential equation f'' = f, then the map psi from R^{2} to S defined by (psi(a_{1}, a_{2}))(t) = a_{1} e^{t} + a_{2} e^{-t} is a linear map.
If S is the vector space of all solutions of the differential equation f'' = f, then the map xi from S to S defined by (xi(f))(t) = f(-t) is a linear map.
Prove that if f is a linear map from V to W, 0_{V} is the zero element of V, and 0_{W} is the zero element of W, then f(0_{V}) = 0_{W}.
For a in R^{n} the map called translation by a is the map T_{a} from R^{n} to R^{n} that is defined by T_{a}(x) = x + a. Is translation by a a linear map?
Compute phi(f_{1}) and phi(f_{2}) when f_{1}(t) = e^{t} and f_{2}(t) = e^{-t} (and phi is the notation introduced in the examples above).
Compute phi( cosh ) and phi( sinh ) where cosh and sinh are the standard hyperbolic functions.
Compute the composition D \circ psi (notation of the examples).
Compute the composition phi \circ psi (notation of the examples).
Find a matrix M such that phi \circ psi = L_{M} (notation of the examples).
In the study of differential equations it is proved that the general solution of the differential equation f'' = f is f(t) = c_{1} e^{t} + c_{2} e^{-t} where c_{1} and c_{2} are arbitrary constants. Rephrase this statement as a statement about the linear map psi.
Show that D(f) is in S if f is in S (notation of the examples). Then find a matrix M such that phi \circ D \circ psi = L_{M}.