Discussion
Let \(A\) be any \(n \times m\) matrix, where \(m \gt n\). Which of the following statements is/are TRUE about the system of linear equations \(Ax = 0\) ?
Answer: A
The true statement is There exist at least \(m - n\) linearly independent solutions to this system.
The rank of A is at most n, its number of rows. By the rank-nullity theorem the solution space of \(Ax = 0\) has dimension \(m - \text{rank}(A) \ge m - n\), so there are at least \(m - n\) linearly independent solutions.
The second statement claims exactly \(m - n\) vectors span every solution, which fails when the rank is less than n; for A = 0 every vector in \(\mathbb{R}^{m}\) is a solution. The third fails in general: a non-zero solution is guaranteed only with at least \(m - n - 1\) zero variables. The fourth fails, for example, for A = [1 0 0; 0 1 0], whose solutions have only the third variable non-zero, fewer than n = 2.
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