If A is a nonsingular matrix, what is the null space of A?
If A is nonsingular then the equation:
A x = y
has a unique solution. Now, the equation:
A x = 0
always has a solution:
x = 0.
If A is nonsingular, then this must thus be the only solution. This means that the null space of A is the zero dimensional vector space containing only the null vector 0.
If A is a nonsingular matrix, the null space of A is the zero-dimensional vector space that only contains the null vector (0). This means that the only solution to the equation Ax = 0 is x = 0.
To understand why the null space of A is the zero dimensional vector space containing only the null vector 0 when A is a nonsingular matrix, we need to understand what the null space represents and how it is related to a nonsingular matrix.
The null space of a matrix A, denoted as null(A) or N(A), is the set of vectors x such that A x = 0. In other words, it is the set of solutions to the homogeneous equation A x = 0.
When A is nonsingular, it means that A has a unique inverse. This implies that for any non-zero vector y, the equation A x = y has a unique solution. However, when A x = 0, it means that the vector y is the zero vector (y = 0).
If A x = y has a unique solution for every non-zero vector y, but A x = 0 has only one solution (x = 0), it means that the null space of A contains only the zero vector 0. This is because no other vector x satisfies the equation A x = 0, except for x = 0.
Therefore, when A is nonsingular, the null space of A is the zero dimensional vector space containing only the null vector 0.