I'm quite confused about what "into" means here and, more importantly, how I am supposed to prove that something maps a vector space into (not onto) another vector space.

Here's some of the background information that I was given:

  • The set V of all solutions x(t) to the homogeneous equation dx/dt = 0 is a vector space of functions.
  • The constant function 1 and the function t, are a basis for the two dimensional vector space V.
  • V consists of all linear polynomials a + bt.
  • The particular solution x(t) to dx/dt = -g with x(0) = 0, (dx/dt)(0) = 0 is (gt/2). Every solution x(t) to dx/dt = -g is the sum of a particular solution to this equation plus a solution from V to the corresponding homogeneous equation dx/dt = 0.
  • If we use D for the operator of differentiation Dx = dx/dt , then D: V --> V, and D is a linear transformation.
  • The original equation can be written Dx = g.
  • If I --> V is the notation for the identity map, this equation can also be written (D I)x = g.
  • The corresponding homogeneous equation is Dx = 0.
  • The vector space V is the set of all x(t) such that Dx = 0, which is the kernel (null space) of D.