Question

Consider the following statements.
P : If a system of linear equations Ax = b has a unique solution, where A is an m × n matrix and b is an m × 1 matrix, then m = n.
Q : For a subspace W of a nonzero vector space V, whenever u ∈ V ∖ W and v ∈ V ∖ W, then u + v ∈ V ∖ 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

Answer 1

The Correct Option is D

Let's evaluate the given statements P and Q based on the principles of Linear Algebra:

1. Statement 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 \(m = n\)." 
   • For a system of linear equations to have a unique solution, the matrix \(A\) must be square (i.e., \(m = n\)) and nonsingular (determinant not equal to zero).
   • However, even if \(m \neq n\), a unique solution is possible if the system is consistent and every variable corresponds to a pivot column in row-echelon form.
   • Hence, the condition \(m = n\) is not a requirement for uniqueness by itself. Thus, statement P is false.
2. Statement Q: "For a subspace \(W\) of a nonzero vector space \(V\), whenever \(u \in V \setminus W\) and \(v \in V \setminus W\), then \(u + v \in V \setminus W\)."
   • If \(u\) and \(v\) are vectors outside the subspace \(W\), their sum \(u + v\) could still belong to the subspace \(W\).
   • For example, if \(W\) is a line in \(\mathbb{R}^2\) and \(u\) and \(v\) are two points on opposite sides of this line, \(u + v\) could be a point on the line \(W\).
   • Therefore, statement Q is false as well because \(u + v\) can sometimes be inside the subspace \(W\).

Based on our analysis:

• Statement P is false as the requirement \(m = n\) is not necessary for the uniqueness of the solution.
• Statement Q is false because the sum of vectors outside a subspace can lie within that subspace.

Conclusion: Both P and Q are false.