Well, you could look at the definitions of those words! Any function, from one set to another is call "injective" if and only if f(x)= f(y) implies x= y: in other words, different members of the domain are mapped to different members of the range. That does NOT imply "surjective"- that something is mapped to every member of the range. For linear transformations, represented by a matrix, injective means that the domain space is mapped one-to-one to a subspace of the range space: that the rank of the matrix is equal to the dimension of the domain space, not necessairily the dimension of the range space.
In order to be "bijective" a mapping must be both injective and surjective: "one-to-one" and "onto". For a linear transformation represented by a matrix, that means it must be n by n for some positive integer n and have rank n.
If f:A-->B is a function, then A is called the domain and the subset f(A) of B is called the range. In the case where f is a linear map from one vector space to another, A is the domain space (or domain vector space) and f(A) is the range space.