Question

Consider the following statements. P: If a system of linear equations \( Ax = b \) has a unique solution, where \( A \) is an \( m \times n \) matrix and \( b \) is an \( m \times 1 \) matrix, then \( m = n \). Q: For a subspace \( W \) of a nonzero vector space \( V \), whenever \( u \in V \setminus W \) and \( v \in V \setminus W \), then \( u + v \in V \setminus W \). Which one of the following holds?

• A. Both P and Q are true
• B. P is true but Q is false
• C. P is false but Q is true
• D. Both P and Q are false

Hint:

To verify such statements, carefully check the conditions and counterexamples where the rules do not hold.

Answer 1

The Correct Option is D

- P: If the system of linear equations \( Ax = b \) has a unique solution, it implies that the matrix \( A \) is invertible, which requires that \( m = n \) (i.e., the number of equations must equal the number of variables). However, it is possible for a system to have a unique solution even if \( m \neq n \), such as when the system is overdetermined but the equations are consistent and independent. Therefore, statement P is false. - Q: Consider a subspace \( W \) of a vector space \( V \). The statement says that if \( u \in V \setminus W \) and \( v \in V \setminus W \), then \( u + v \in V \setminus W \). This is not necessarily true. For example, if \( u \) and \( v \) are both outside \( W \), their sum may belong to \( W \). Hence, statement Q is false. Thus, the correct answer is (D): Both P and Q are false.