Question

Let \(V\) be a vector space over a field \(F\) and \(a \in F\) and \(u \in V\). Which of the following statements is \textbf{not correct?}

• A. \(au = \theta \Rightarrow\) either \(a = 0\) or \(u = \theta\)
• B. \(|-1\,u| = |-1|\,|u|\) for all \(u \in V\)
• C. \(a\theta = \theta\)
• D. \(0u = \theta \;\; a \in F\)

Hint:

Remember:

(0u = theta) depends only on the scalar (0), not on any arbitrary (a in F)
Always check statements for unnecessary or incorrect conditions

Answer 1

The Correct Option is D

Step 1: Examine option (A). In a vector space, \[ au = \theta \Rightarrow a = 0 \text{ or } u = \theta \] This is a standard property of vector spaces. Hence, (A) is correct. Step 2: Examine option (B). \[ |-1\,u| = |-1|\,|u| = 1\cdot |u| = |u| \] This follows from the norm property: \[ |cu| = |c|\,|u| \] So, (B) is correct. Step 3: Examine option (C). Scalar multiplication of the zero vector always gives the zero vector: \[ a\theta = \theta \] Hence, (C) is correct. Step 4: Examine option (D). The correct vector space property is: \[ 0u = \theta \] which is true independent of any scalar \(a\). However, option (D) incorrectly includes the condition “\(a \in F\)”, which is irrelevant and incorrect in this context. Step 5: Final conclusion. Thus, the statement which is not correct as written is option (D).