Final answer:
The null space of a matrix is the set of all vectors that when multiplied by the matrix result in the zero vector. For matrices A, B, and C, the corresponding homogeneous systems of linear equations are created by equating the matrix-vector product to zero, resulting in the null space for each matrix.
Step-by-step explanation:
To find the null space of a matrix, we equate each row of the matrix to zero. This gives us a homogeneous system of linear equations that corresponds to the null space. Let's do this for each matrix given.
Matrix A
Matrix A = [1 -2][2 -4]. The homogeneous system that corresponds to the null space of matrix A is obtained by applying the matrix to a vector (x, y) and setting it to the zero vector (0, 0).
1*x - 2*y = 0
2*x - 4*y = 0
Matrix B
Matrix B = [1 -3 0 1][2 5 -1 0][3 2 -1 1]. The homogeneous system for matrix B would be:
1*x - 3*y + 0*z + 1*w = 0
2*x + 5*y - 1*z + 0*w = 0
3*x + 2*y - 1*z + 1*w = 0
Matrix C
Matrix C = [3 1][-4 -1]. The homogeneous system for matrix C is:
3*x + 1*y = 0
-4*x - 1*y = 0
The solutions to these equations make up the null space or kernel of the matrices which is a set of all vectors that when multiplied by the matrix yield the zero vector.