Chapter 2.1, Problem 43E

a. Consider the vector $\stackrel{\to }{v}=\left[\begin{array}{l}2\\ 3\\ 4\end{array}\right]$ . Is the transformation $T\left(\stackrel{\to }{x}\right)=\stackrel{\to }{v}\cdot \stackrel{\to }{x}$ (the dot product) from ${ℝ}^3$ to $ℝ$ linear?If so, find the matrix of T.
b. Consider an arbitrary vector $\stackrel{\to }{v}$ in ${ℝ}^3$ . Is the transformation $T\left(\stackrel{\to }{x}\right)=\stackrel{\to }{v}\cdot \stackrel{\to }{x}$ ? linear? If so, find the matrix ofT (in terms of the components of $\stackrel{\to }{v}$ ).
c. Conversely, consider a linear transformation T from ${ℝ}^3$ to $ℝ$ . Show that there exists a vector $\stackrel{\to }{v}$ in ${ℝ}^3$ suchthat $T\left(\stackrel{\to }{x}\right)=\stackrel{\to }{v}\cdot \stackrel{\to }{x}$ , for all $\stackrel{\to }{x}$ in ${ℝ}^3$ .