Definition of the column space of a matrix; column space is a subspace; comparison to the null space; definition of a linear transformation between vector spaces; definition of kernel and range of a linear transformation
Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. We will see that they are closely related to ideas like linear independence and spanning, and subspaces like the null space and the column space. In this section we will define an injective linear transformation and analyze the resulting consequences. The next section will do the same for the surjective property. In the final section of this chapter we will see what happens when we have the two properties simultaneously.