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: \[ \boxed{(1)} \]