91Ó°ÊÓ

Let \(V\) and \(V^{\prime}\) be vector spaces over the same field \(F, A\) function \(\phi: V \rightarrow V^{\prime}\) is a linear transformation of \(V\) ) into \(V^{\prime}\) if the following conditions are satisfied for all \(\alpha, \beta \in V\) and \(a \in F\) : $$ \begin{aligned} \phi(\alpha+\beta) &=\phi(\alpha)+\phi(\beta) \\ \phi(a \alpha) &=a(\phi(\alpha)) \end{aligned} $$ a. If \(\left\\{\beta_{i} \mid i \in I\right\\}\) is a basis for \(V\) over \(F\), show that alinear transformation \(\phi: V \rightarrow V^{\prime}\) is completely determined by the vectors \(\phi\left(\beta_{i}\right) \in V^{\prime}\). b. Let \(\left(\beta_{1} \mid i \in I\right)\) be a basis for \(V\), and let \(\left\\{\beta_{i}^{\prime} \mid i \in I\right\\}\) be any set of vectors, not necessarily distinct, of \(V^{\prime}\). Show that there exists exactly one linear transformation \(\phi: V \rightarrow V^{\prime}\) such that \(\phi\left(\beta_{i}\right)=\beta_{i}^{\prime}\).

Let \(V\) and \(V^{\prime}\) be vector spaces over the same field \(F\), and let \(\phi: V \rightarrow V^{\prime}\) be a linear transformation. a. To what concept that we have studied for the algebraic structures of groups and rings does the concept of a linear transformation correspond? b. Define the kemel (or nullspace) of \(\phi\), and show that it is a subspace of \(V\). c. Describe when \(\phi\) is an isomorphism of \(V\) with \(V^{\prime}\).

Let \(V\) be a vector space over a field \(F\), and let \(S\) be a subspace of \(V\). Define the quotient space \(V / S\), and show that it is a vector space over \(F\).