Kernel and Image of a Linear Map: Rank-Nullity Theorem

The kernel of a linear map T is the set of inputs T sends to zero; the image is the set of outputs T actually produces. The kernel measures collapse, how many independent directions flatten to nothing, while the image measures reach, the subspace of the target genuinely hit.

They are linked by the rank-nullity theorem: dimension lost to the kernel plus dimension surviving in the image always equals the starting dimension. So a map from R cubed onto R squared must crush at least one direction. It underlies every criterion for when linear systems have solutions and when they are unique.

Key Concepts

• Kernel ker(T) = (v : T(v) = 0), a subspace of the domain; T is injective exactly when it is (0).
• Image im(T) = (T(v) : v in V), a subspace of the codomain; T is surjective exactly when it is the whole codomain.
• Rank: dim(im T), the number of pivot columns. Nullity: dim(ker T), the number of free variables.
• Rank-nullity theorem: rank(T) + nullity(T) = dim(domain).

Formulas

• Rank-nullity: dim(ker T) + dim(im T) = dim(domain).
• Injectivity: nullity = 0, equivalently independent columns.
• Surjectivity: rank = dim(codomain), equivalently spanning columns.
• Square case: on R to the n, injective, surjective and invertible coincide, each equivalent to nonzero determinant.
• Solvability: Ax = b is solvable exactly when b is in the image of A.

Worked Example

Let T(x,y,z) = (x+y, y+z) from R cubed to R squared, matrix [[1,1,0],[0,1,1]]. Kernel: x+y = 0 and y+z = 0 give x = minus y, z = minus y; with y = t the solutions are t(minus 1,1,minus 1), so nullity = 1. Image: the columns (1,0), (1,1), (0,1) include e1 and e2, spanning all of R squared, so rank = 2. Check: 2 + 1 = 3 = dim(R cubed). One direction collapses, two survive, exactly as the theorem demands.

Common Misconceptions

• Nonzero kernel means useless map. It means some information is intentionally discarded, like depth in a projection; the image still delivers the design goal.
• Rank counts nonzero rows. Rank is the number of pivot columns. Three nonzero rows can still have rank 1 if all multiples of each other.
• Rank-nullity works in infinite dimensions. As stated it needs a finite-dimensional domain.

Real-World Applications

• Linear systems: the general solution of Ax = b is a particular solution plus the whole kernel.
• Computer vision: a camera projection has 1D kernel (depth along sight rays) and 2D image (the photo).
• Differential equations: solutions of a homogeneous linear ODE form the kernel of a differential operator.

Kernel and image split the domain into lost and surviving parts, and rank-nullity keeps the books: rank plus nullity equals domain dimension. Compute one side from the matrix, get the other free, and read off injectivity and surjectivity.