To find the null space of a matrix, we need to solve the homogeneous system of linear equations given by Ax = 0, where A is the matrix and x is a vector of variables. The null space, also known as the kernel, consists of all possible solutions to this system.

  There are several methods to find the null space of a matrix, such as Gaussian elimination and the row echelon form method. Here is a step-by-step process using Gaussian elimination:

  1. Write the augmented matrix [A | 0], where A is the given matrix and the column of zeros represents the right-hand side of the equation.

  2. Perform row operations to transform the matrix into row echelon form or reduced row echelon form.

   - Start by choosing a pivot element in the first column and row (usually the leftmost non-zero entry).

   - Eliminate or reduce all non-zero elements below the pivot to zeros by subtracting multiples of the pivot row from rows below it.

   - Move to the next column and repeat the process, taking care to choose a pivot element below the previous pivot element.

  3. Once the matrix is in row echelon form or reduced row echelon form, the null space can be determined.

   - If there is a row of all zeros, it means the system has***** variables, and the null space will be infinite-dimensional.

   - If there is no row of all zeros, the null space will be zero-dimensional, consisting of only the trivial solution.

  4. Express the solution set in parametric form if there are***** variables. Each***** variable corresponds to a parameter, and the solution set is expressed as a linear combination of these parameters.

  It is important to note that finding the null space of a matrix is closely related to the concept of linear independence and dependence of the column vectors. If the matrix has linearly independent columns, the null space will only contain the zero vector.