At the center of almost any proof involving linear transformations is to apply linearity to move between the domain and codomain, preserving the structure of linear combinations. Statements proved in such a way:

• all linear transformations take 0 to 0
• images of subspaces are subspaces (special example: image)
• preimages of subspaces are subspaces (special example: kernel)
• closing a set of vectors before or after applying a transformation gives the same vector space
• we may unambiguously determine the entirety of a linear transformation from its action on a basis (linear extension) — which is what makes matrix representation possible!
• having an inverse is equivalent to being bijective
• the rank of a matrix is definable by its rows as well as its columns
• inverses of isometries are also isometries (also uses bilinearity of inner product)