Question

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 $Ax = 0$?

• A. There exist at least $m - n$ linearly independent solutions to this system
• B. There exist $m - n$ linearly independent vectors such that every solution is a linear combination of these vectors
• C. There exists a non-zero solution in which at least $m - n$ variables are 0
• D. There exists a solution in which at least $n$ variables are non-zero

Hint:

For underdetermined systems (( m>n )), the null space dimension is key to determining the number of linearly independent solutions.

Answer 1

The Correct Option is A

Step 1: Analyze the null space of $A$.
For an $n \times m$ matrix $A$, the rank-nullity theorem states: $$\text{Nullity}(A) = m - \text{Rank}(A).$$ Given $m>n$, the rank of $A$ is at most $n$, so the nullity is at least $m - n$. Step 2: Validate the options.
Option (1): There are at least $m - n$ linearly independent solutions in the null space of $A$, which is correct.
Option (2): While $m - n$ vectors span the null space, this does not imply they are solutions to $Ax = 0$. Incorrect.
Option (3): There is no guarantee that $m - n$ variables in the solution are 0. Incorrect.
Option (4): There is no guarantee that at least $n$ variables are non-zero in the solution. Incorrect. Final Answer: $$\boxed{(1)}$$