Kernel, range, and invertibility
Two subspaces describe what a linear map loses and what it reaches: the kernel and the range. They determine whether the map is one-to-one, onto, or invertible.
Kernel
The kernel of a linear map T is the set of vectors sent to zero: \(\ker T = \{\mathbf{x} : T(\mathbf{x}) = \mathbf{0}\}\). For a matrix transformation, the kernel is the null space of A. The kernel is always a subspace of the domain, and it measures what information the map destroys.
Range
The range of T is the set of all outputs T(x), a subspace of the codomain. For a matrix transformation it is the column space of A. Its dimension is the rank of T. The range measures what the map can reach.
One-to-one maps
T is one-to-one (injective) if different inputs always give different outputs. For linear maps this is equivalent to \(\ker T = \{\mathbf{0}\}\): if \(T(\mathbf{u}) = T(\mathbf{v})\), then \(T(\mathbf{u} - \mathbf{v}) = \mathbf{0}\), so u - v is in the kernel. For matrices, one-to-one means independent columns, a pivot in every column.
Onto maps
T is onto (surjective) if every vector in the codomain is an output. For a matrix transformation from \(\mathbb{R}^n\) to \(\mathbb{R}^m\), onto means the columns span \(\mathbb{R}^m\), a pivot in every row. A map from \(\mathbb{R}^3\) to \(\mathbb{R}^2\) can be onto but never one-to-one; one from \(\mathbb{R}^2\) to \(\mathbb{R}^3\) can be one-to-one but never onto.
Rank-nullity for maps
For a linear map T from a finite-dimensional space V, dim ker T + dim range T = dim V. This is the rank-nullity theorem stated for transformations: dimensions that are lost plus dimensions that survive add up to the dimension of the input space.
Invertible maps and isomorphisms
A linear map that is both one-to-one and onto is invertible: its inverse is also linear. For T from \(\mathbb{R}^n\) to \(\mathbb{R}^n\) with matrix A, invertibility of T is equivalent to invertibility of A, and the inverse map has matrix \(A^{-1}\). An invertible linear map between spaces is an isomorphism; spaces are isomorphic exactly when they have the same dimension.
Square matrices: one implies the other
For a linear map from \(\mathbb{R}^n\) to itself, one-to-one and onto are equivalent: by rank-nullity, a trivial kernel forces the range to have dimension n. So for square matrices, independence of columns, spanning, and invertibility all go together, the heart of the invertible matrix theorem.
Worked example
The differentiation map D from polynomials of degree at most 3 to polynomials of degree at most 2 sends \(a + bt + ct^2 + dt^3\) to \(b + 2ct + 3dt^2\). Its kernel is the constant polynomials, dimension 1, so D is not one-to-one. Its range is all quadratics, dimension 3, so D is onto. Rank-nullity checks: 1 + 3 = 4, the dimension of the domain.
Key takeaways
The kernel, vectors mapped to zero, is a subspace of the domain (the null space for matrices). The range, all outputs, is a subspace of the codomain (the column space), with dimension the rank. Linear maps are one-to-one exactly when the kernel is trivial (pivot in every column) and onto when the range is the whole codomain (pivot in every row). dim ker T + dim range T = dim V. Invertible maps are one-to-one and onto, with linear inverses; isomorphic spaces have equal dimension. For maps from R^n to R^n, one-to-one, onto, and invertible are equivalent.
Flashcard
Vectors mapped to zero
Check your understanding
A linear map T from R^5 to R^3 is onto. What is the dimension of its kernel?