## Prove that D (the differential operator) maps V (a vector space) into V

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 d²x/dt² = 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 d²x/dt² = -g with x(0) = 0, (dx/dt)(0) = 0 is –(gt²/2). Every solution x(t) to d²x/dt² = -g is the sum of a particular solution to this equation plus a solution from V to the corresponding homogeneous equation d²x/dt² = 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 D²x = –g.
• If I --> V is the notation for the identity map, this equation can also be written (D² – I)x = –g.
• The corresponding homogeneous equation is D²x = 0.
• The vector space V is the set of all x(t) such that D²x = 0, which is the kernel (null space) of D².