kernel and image (coordinate-free)
In Volume I the kernel was the null space of a matrix and the image was its column space — both computed by row-reducing. The coordinate-free view drops the matrix entirely. For a linear map T : V -> W, the kernel is the set of vectors V sends to zero, ker T = { v in V : T(v) = 0 }, and the image is the set of vectors T actually reaches, im T = { T(v) : v in V }.
Each is automatically a subspace: the kernel is a subspace of the domain V, the image is a subspace of the codomain W. You verify this directly from linearity — if T(u) = 0 and T(v) = 0 then T(u + v) = 0, and similarly for image — with no basis, no matrix, no row operations in sight. These subspaces are intrinsic to the map: they do not change when you change basis.
This intrinsic-ness is the whole point. The matrix of T depends on a choice of bases and changes when you change them; but the kernel and image are honest geometric objects attached to T alone. Their dimensions — the nullity and the rank — are therefore invariants of the map, the same in every coordinate system. Row-reducing a matrix is just one way to compute objects that exist before any matrix does.
Two clean criteria live here. T is injective exactly when ker T = { 0 } (nothing nonzero is crushed), and T is surjective exactly when im T = W (everything is reached). Holding both subspaces in mind, rather than a matrix, is the habit that makes the isomorphism theorems feel inevitable.
Both fundamental properties of a map read off directly from its two intrinsic subspaces, with no matrix needed.
Kernel lives in the domain, image lives in the codomain — they are subspaces of different spaces and should not be added or compared directly. The first isomorphism theorem is precisely what relates them, via the quotient V/ker T.