Define the null space of a matrix.

The null space of a matrix is the set of all solutions to the homogeneous equation Ax=0.

When we multiply a matrix A by a vector x, we get a new vector Ax. The null space of A is the set of all vectors x that satisfy the equation Ax=0. In other words, the null space is the set of all solutions to the homogeneous equation Ax=0.

To find the null space of a matrix, we can use Gaussian elimination to row-reduce the augmented matrix [A|0]. The null space will then be the set of all solutions to the resulting system of linear equations. We can also use the fact that the null space is the orthogonal complement of the row space of A.

The null space is also known as the kernel of A. It is a subspace of the vector space R^n, where n is the number of columns in A. The dimension of the null space is called the nullity of A, and it can be found by counting the number of free variables in the row-reduced form of [A|0].

The null space is an important concept in linear algebra, as it allows us to understand the structure of the solutions to linear systems of equations. It is also used in applications such as image compression and data analysis.