91Ó°ÊÓ

Let \(\mathrm{W}\) be a subspace of a vector space \(\mathrm{V}\) over a field \(F\). For any \(v \in \mathrm{V}\) the set \(\\{v\\}+\mathrm{W}=\\{v+w: w \in \mathrm{W}\\}\) is called the coset of \(\mathrm{W}\) containing \(v\). It is customary to denote this coset by \(v+\mathrm{W}\) rather than \(\\{v\\}+\mathrm{W}\). (a) Prove that \(v+\mathrm{W}\) is a subspace of \(\mathrm{V}\) if and only if \(v \in \mathrm{W}\). (b) Prove that \(v_{1}+\mathrm{W}=v_{2}+\mathrm{W}\) if and only if \(v_{1}-v_{2} \in \mathrm{W}\). Addition and scalar multiplication by scalars of \(F\) can be defined in the collection \(S=\\{v+\mathrm{W}: v \in \mathrm{V}\\}\) of all cosets of \(\mathrm{W}\) as follows: $$ \left(v_{1}+\mathbf{W}\right)+\left(v_{2}+\mathbf{W}\right)=\left(v_{1}+v_{2}\right)+\mathbf{W} $$ for all \(v_{1}, v_{2} \in \mathrm{V}\) and $$ a(v+\mathrm{W})=a v+\mathrm{W} $$ for all \(v \in \mathrm{V}\) and \(a \in F\). (c) Prove that the preceding operations are well defined; that is, show that if \(v_{1}+\mathrm{W}=v_{1}^{\prime}+\mathrm{W}\) and \(v_{2}+\mathrm{W}=v_{2}^{\prime}+\mathrm{W}\), then $$ \left(v_{1}+\mathbf{W}\right)+\left(v_{2}+\mathbf{W}\right)=\left(v_{1}^{\prime}+\mathbf{W}\right)+\left(v_{2}^{\prime}+\mathbf{W}\right) $$ and $$ a\left(v_{1}+\mathrm{W}\right)=a\left(v_{1}^{\prime}+\mathrm{W}\right) $$ for all \(a \in F\). (d) Prove that the set \(S\) is a vector space with the operations defined in (c). This vector space is called the quotient space \(V\) modulo \(\mathrm{W}\) and is denoted by \(\mathrm{V} / \mathrm{W}\).