91Ó°ÊÓ

First, we need to show that if \(W\) is a subspace of \(V\), then \(0 \in W\) and \(ax + y \in W\) whenever \(a \in F\) and \(x, y \in W\). By definition, for \(W\) to be a subspace of \(V\), it needs to satisfy three conditions: 1. The zero vector (denoted as \(0\)) is in \(W\). 2. For all \(x, y \in W\), also \(x + y \in W\). 3. For all \(x \in W\) and \(a \in F\), where F is the field of scalar coefficients, also \(ax \in W\). Since we are already given that \(W\) is a subspace of \(V\), condition 1 is trivially satisfied, which means \(0 \in W\). Now, we are asked to prove that \(a x + y \in W\) whenever \(a \in F\) and \(x, y \in W\). We know that \(x \in W\) and \(y \in W\), so we can use condition 2, which states that \(x + y \in W\). Similarly, since \(a \in F\) and \(x \in W\), condition 3 tells us that \(ax \in W\). Now, as \(ax \in W\) and \(y \in W\), we can apply condition 2 again, giving us that \(ax + y \in W\). Therefore, we have shown that if \(W\) is a subspace of vector space \(V\), then \(0 \in W\) and \(ax + y \in W\) whenever \(a \in F\) and \(x, y \in W\).

Now, we will show that if \(0 \in W\) and \(ax + y \in W\) whenever \(a \in F\) and \(x, y \in W\), then \(W\) is a subspace of \(V\). We must show that the three conditions for a subspace are satisfied. 1. We are already given that \(0 \in W\). Thus, condition 1 is satisfied. 2. We need to show that for all \(x, y \in W\), also \(x + y \in W\). We can rewrite this as \(1x + y \in W\). Since \(1 \in F\), this property holds due to the given condition. 3. We need to show that for all \(x \in W\) and \(a \in F\), also \(ax \in W\). This is already given to us as part of the hypothesis. Hence, all three conditions of a subspace are satisfied, and we can conclude that \(W\) is indeed a subspace of \(V\). In summary, we have shown that a subset \(W\) of a vector space \(V\) is a subspace of \(V\) if and only if \(0 \in W\) and \(ax + y \in W\) whenever \(a \in F\) and \(x, y \in W\).