The kernel (or nullspace) of a linear transformation is the set of vectors such that It is a subspace of whose dimension is called the nullity. The rank-nullity theorem relates this dimension to the rank of
When is given by left multiplication by an matrix so that where is thought of as an matrix it is common to refer to the kernel of the matrix rather than the kernel of the linear transformation, i.e. instead of
Many subspaces of can be naturally described as kernels of a particular linear transformation (and every subspace of can be described as the kernel of some linear transformation). Given a system of linear equations the computation of the kernel of (via Gaussian elimination) can be used to give a general solution to the system once a particular solution is known.
Gaussian elimination: It is a fact (proved in the below section) that row reduction doesn't change the kernel of a matrix. The kernel of the matrix at the end of the elimination process, which is in reduced row echelon form, is computed by writing the pivot variables ( in this case) in terms of the free (non-pivot) variables ( in this case). That is, is equivalent to
so the solution vector equals
So the kernel is a one-dimensional subspace of whose basis is the single vector
- Injectivity: The kernel gives a quick check on the injectivity of :
A linear transformation is injective if and only if
To see this, note that the kernel is the set of vectors which map to , so if is injective then the kernel can only have one element, which must be . On the other hand, if the kernel is trivial, then implies that , and since the kernel is trivial, this implies so So is injective.
Subspace: To see that the kernel is a subspace of it is enough to check that it is closed under addition and scalar multiplication:
This straightforward result is proved in the wiki on row and column spaces.
Unchanged under elementary row operations: As cited in the above example, the elementary row operations in Gaussian elimination do not change the kernel of . One way to see this is that doing an elementary row operation is the same as multiplying on the left by an invertible elementary matrix But the kernel of is the same as the kernel of : if then and if then so This justifies the method for computing the kernel outlined in the above example.
Solving general linear equations via translation: As is common with linear systems of equations, the kernel can be used to solve general equations of the form If then the general solution is of the form where This is because
So the set of solutions to is a coset of the kernel, of the form
For find all solutions to given that is one such solution.
The example in the introduction shows that the kernel of is the set of vectors which are scalar multiples of so the set of all solutions to is for all
Intuitively, the kernel measures how much the linear transformation collapses the domain If the kernel is trivial, so that does not collapse the domain, then is injective (as shown in the previous section); so embeds into But if the kernel is nontrivial, is no longer an embedding, so its image in is smaller.
This intuition suggests an inverse relationship between the sizes of the kernel and of the image of The formal version of this intuition is the rank-nullity theorem. Here it is stated in matrix form:
Here the rank of is the dimension of the column space (or row space) of The first term of the sum, the dimension of the kernel of is often called the nullity of
Let be an matrix. Then
The most natural way to see that this theorem is true is to view it in the context of the example from the previous two sections. The computation of the kernel of made it clear that the dimension of the kernel was equal to the number of free (non-pivot) columns in the reduced row echelon form of On the other hand, recall that the rank equals the number of pivot columns. So the sum of the dimensions equals the number of columns, which is
Again taking verify rank-nullity for
We have already seen that and the rank of equals the number of pivot columns in the reduced row echelon form which is As expected,
This wiki has concentrated on down-to-earth applications of the kernel using matrices, but the statements above hold more generally for linear transformations between arbitrary vector spaces The kernel is still a subspace and can still be used to solve linear equations of the form the rank-nullity theorem is still correct if the "number of columns" is replaced by
Let the real vector space of polynomials of degree with real coefficients. Verify rank-nullity for the linear transformation given by
The kernel of is the set of polynomials in whose second derivative vanishes. This is clearly the set of polynomials of the form with This is a two-dimensional subspace of with basis
The image of is the set of polynomials in which are the second derivative of a polynomial in It is not hard to see that this is the subspace of consisting of polynomials of degree This is a three-dimensional subspace of with basis
The dimension of is since its canonical basis is Rank-nullity is satisfied since