Final answer:
The standard method for producing a spanning set and subsequently a basis for the null space of matrix A is reliable and does not fail when applied correctly. The resulting set of vectors are linearly independent and span the null space, thereby forming a basis. The correct answer to the student's question is B. False.
Step-by-step explanation:
The question is related to the topic of linear algebra, specifically the null space (nullity) of a matrix, denoted as nul A. The null space of a matrix A consists of all the vectors that when multiplied by A result in the zero vector. To find a spanning set for the null space of matrix A (nul A), one typically performs the Gaussian elimination or row reduction to bring the matrix to its reduced row-echelon form (RREF). From the RREF, a set of free variables is determined which can be assigned parameters to express the basic variables in terms of these free variables. The set of vectors formed in this process spans the null space of A.
However, the answer to the question, 'The standard method for producing a spanning set for nul A sometimes fails to produce a basis for nul A' is B. False. The vectors obtained through the standard method indeed form a basis for the null space if the method has been correctly applied. This basis consists of linearly independent vectors that span the null space, fulfilling both criteria needed for the set to be a basis. The confusion may arise if the reduction to RREF is not performed completely or correctly, leading to an erroneous conclusion that the method 'fails'. However, when the procedure is carried out accurately, the resulting set of vectors representing the solutions to the homogeneous system Ax = 0 will invariably form a basis for nul A.