Let $A$ be any $n \times m$ matrix, where $m > n$. Which of the following statements is/are TRUE about the system of linear equations $Ax = \mathbf{0}$ ?

For a homogeneous linear system $Ax = \mathbf{0}$ where $A$ is an $n \times m$ matrix:

1. By the Rank-Nullity Theorem, the dimension of the solution space (null space of $A$) is:

   $$\text{nullity}(A) = m - \text{rank}(A)$$

   Since $\text{rank}(A) \le \min(n, m) = n$, we have:

   $$\text{nullity}(A) \ge m - n$$

   Therefore, the null space has dimension at least $m - n$, which means there exist at least $m - n$ linearly independent solutions. Hence, (A) is TRUE.

2. Statement (B) claims there exist *exactly* $m - n$ linearly independent vectors that span the solution space. If $\text{rank}(A) < n$, the dimension of the null space is strictly greater than $m - n$, so $m - n$ vectors cannot span the entire solution space. Hence, (B) is FALSE.

3. Counterexample for (C): Consider $n = 1, m = 2$, and $A = [1 \quad 1]$. Then $m - n = 1$. The equation is $x_1 + x_2 = 0 \implies x_2 = -x_1$. Any non-zero solution is of the form $(c, -c)$ with $c

e 0$, where neither variable is$0$. Thus, there is no non-zero solution with at least$1$ variable being zero. Hence, (C) is FALSE.

4. Counterexample for (D): Consider $n = 2, m = 3$, and $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$. The system $Ax = \mathbf{0}$ yields $x_1 = 0$ and $x_2 = 0$, so any solution is $[0, 0, c]^T$. In every solution, at most $1$ variable is non-zero, so there is no solution with at least $n = 2$ non-zero variables. Hence, (D) is FALSE.

Thus, only A is TRUE.