Let 𝐴 be any 𝑛 × 𝑚 matrix, where 𝑚 > 𝑛. Which of the following statements…

Key idea: Use the rank-nullity theorem.

By the rank-nullity theorem, rank(A) + nullity(A) = m. Since rank(A) ≤ n, we have nullity(A) = m − rank(A) ≥ m − n. The nullspace therefore has dimension at least m − n, so there exist at least m − n linearly independent solutions to A x = 0.

• Statement that there exist at least m − n linearly independent solutions: true. Reason: nullity(A) ≥ m − n, so any basis of the nullspace provides at least m − n independent solutions.

• Statement that there exist m − n linearly independent vectors that span all solutions: false in general. Reason: nullity may be larger than m − n, so you would need more than m − n vectors to span the nullspace. Example: if A is the zero matrix then nullspace = R^m has dimension m, which can exceed m − n.

• Statement that there exists a non-zero solution with at least m − n variables equal to zero: false in general. Counterexample: let n = 2, m = 3 and A = [[1,0,1],[0,1,1]]. Solutions are scalar multiples of (−1,−1,1), which has no zero entries, so there is no nonzero solution with at least m − n = 1 zero.

• Statement that there exists a solution with at least n non-zero variables: false in general. Counterexample: choose A = [I_n | 0_{n×(m−n)}]. Then every solution has first n entries zero and at most m − n nonzeros. If m − n < n (for example m = n + 1), no solution can have n nonzeros.

Conclusion: Only the first statement is guaranteed to be true for every n × m matrix with m > n.