Final answer:
To find the basis for the nullspace of matrix A, the homogeneous equation Ax = 0 is solved to find that vector (1/4, 1) is a basis for the nullspace.
Step-by-step explanation:
To find a basis for the nullspace of the matrix A = | 4 -1 | | -12 3 |, you must solve the homogeneous equation Ax = 0. This involves setting up the equation 4x - y = 0 and -12x + 3y = 0, which simplifies to one equation, as the second is just a multiple of the first. This means we have a system with infinitely many solutions, which can be expressed as multiples of a particular vector. By inspection, it is clear that the vector (1/4, 1) is a solution to our equation since multiplying it by our matrix A yields the null vector with all components being zero. Therefore, we can say that the set containing the vector {(1/4, 1)} is a basis for the nullspace of A.