Final answer:
To find a generating set for the null space of a matrix, solve the homogeneous system of equations represented by the matrix equation Ax = 0 by reducing the augmented matrix to row-echelon form. Identify the free variables and express the dependent variables in terms of the free variables. Write the solutions as linear combinations of vectors with the free variables as coefficients to obtain a generating set for the null space.
Step-by-step explanation:
The null space of a matrix, also known as the kernel, is the set of all vectors that, when multiplied by the matrix, equal the zero vector. To find a generating set for the null space, we need to solve the homogeneous system of equations represented by the matrix equation Ax = 0. Here are the steps:
- Write the augmented matrix [A|0] of the system of equations.
- Perform row operations to reduce the matrix to row-echelon form.
- Identify the free variables by looking for columns without a leading one. These variables can take any value.
- Express the dependent variables in terms of the free variables to find the solution set for the system of equations.
- Write the solutions as linear combinations of vectors with the free variables as coefficients.
The vectors in the null space form a generating set for the null space of the matrix.