Null space
From Exampleproblems
In mathematics, the null space (also nullspace) of an operator A is the set of all operands v which solve the equation Av = 0. It is also called the kernel of A. In set-builder notation,
If A is a matrix, this is a linear subspace of the space of all vectors. The dimension of this linear subspace is called the nullity of A. This can be calculated as the number of columns that don't contain pivots in the row echelon form of the matrix A. The rank-nullity theorem states that the rank of any matrix plus its nullity equals the number of columns of that matrix.
The right singular vectors of A corresponding to zero singular values form a basis for the null space of A.
The null space of A can be used to find and express all solutions (the complete solution) of the equation Ax = b. If x1 solves this equations it is called a particular solution. The complete solution of the equation is equal to the particular solution added to any vectors from the null space. The particular solutions vary according to b, while the null space vectors do not.
To show this works we consider each direction. In one direction, if Ay = b, and Av = 0, then it is clear that A(y+v) = Ay+Av = b+0 = b. So y+v is also a solution of Ax=b. In the other direction, if we have another solution z to Ax=b, then A(z−y) = Az−Ay = b−b = 0. Thus the vector u = z−y is in the null space of A and z = y+u. So any other solution can be found by adding a vector from the null space to the single particular solution y.
A linear mapping A is an isomorphism if and only if its null space is zero. This is so since conversely, if the null space is nonzero, the map is not one-to-one. The identity map is an example of such A.
If the map is a zero map, then the null space is the same as the domain of the map.
Do not confuse the null space with the zero vector space, which is a space of the only zero vector.it:Spazio nullo ja:零空間
