Let A be any m x n matrix, where m < n. Which of the following statements is/are TRUE about the system of linear equations Ax = 0? A. There exist at least n - m linearly independent solutions to this system. B. There exist m linearly independent vectors such that every solution is a linear combination of these vectors. C. There exist M in R^{n x p}, p in R^+, such that M x = p has solutions and M x = q has no solution for q in R^p. D. There exist n - m vectors in R^n such that at least n - m variables are non-zero.

GATE 2024 · Engineering Mathematics · System of Equations · medium

Answer: Only statement A is TRUE: there exist at least n - m linearly independent solutions to Ax = 0.

1. Apply rank-nullity theorem: Since A is m x n, rank(A) <= min(m, n) = m. Therefore nullity(A) = n - rank(A) >= n - m.
2. Verify Statement A: The null space has dimension >= n - m, so there are at least n - m linearly independent vectors in the null space. Statement A is TRUE.
3. Check Statement B: B claims m vectors span all solutions. But nullity = n - rank(A) is not generally equal to m. For example A is 2 x 5 with rank 2: null space dimension = 3 != m = 2. Statement B is FALSE in general.
4. Check Statements C and D: Statement C describes existence of a matrix M and vectors p, q -- this is about a different system and does not follow necessarily. Statement D claims n - m vectors exist with at least n - m nonzero variables; this is too specific and not generally guaranteed from the given information. Both C and D are not universally true.