How can one find a basis for the null space of a matrix?

To find a basis for the null space of a matrix, the procedure involves solving the equation ( Ax = 0 ), where ( A ) is the matrix in question and ( x ) is a vector from the vector space that the matrix operates on. The null space is defined as the set of all vectors ( x ) that satisfy this equation, meaning they produce the zero vector when multiplied by the matrix ( A ).

To determine a basis for this null space, one typically performs row reduction on the matrix to bring it into a simplified form, such as reduced row echelon form. This process allows you to clearly see the solutions to the equation ( Ax = 0 ) by identifying the free variables and forming the general solution. From this general solution, you can extract the specific vectors that form a basis for the null space.

The other options, while relevant to certain aspects of linear algebra, do not directly lead to finding the null space basis. Eigenvalues pertain to the characteristic behavior of the matrix with respect to transformations, while evaluating the characteristic polynomial and determining row echelon form relate more to properties of the matrix rather than directly yielding the null space. Thus, the most direct and correct approach to find a basis for