Chapter 4.3, Problem 72E

In all parts of this problem, let V be the set of all vectors
$\stackrel{\to }{x}$ in ${ℝ}^4$ such that $x_3=x_1+x_2$ and $x_4=x_2+x_3$ .

a. Represent V as the kernel of a matrix M. Find the rank of M and the dimension of V. Show that $\mathfrak{A}=\left(\left[\begin{array}{l}1\\ 0\\ 1\\ 1\end{array}\right],\left[\begin{array}{l}0\\ 1\\ 1\\ 2\end{array}\right]\right)$ , is a basis of V.

b. Find all vectors of the form $\left[\begin{array}{l}1\\ r\\ r^2\\ r^3\end{array}\right]$

that are contained in V. (Be prepared to deal with irrational numbers.) Can you form a basis of V consisting of such vectors?

c. Consider the linear transformation $T\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{l}{x}_2\\ x_3\\ x_4\\ x_3+x_4\end{array}\right]$ from ${ℝ}^4$ to ${ℝ}^4$.If $\stackrel{\to }{x}$ is a vector in V, show that $T\left(\stackrel{\to }{x}\right)$ is in V as well. Thus, T induces a linear transformation from V to V, which we will denote by F.
d. Find the matrix A of F with respect to the basis $\mathfrak{A}$ from part (a). [Note that A will be a $2\times 2$ matrix, since $\dim \left(V\right)=2$ .]

e. Find the matrix B of F with respect to your basis from part (b).

f. Find the change of basis matrix
g. Write an equation relating the matrices A, B, and S, and check that this equation holds for the matrices you found in parts (d), (e), and (f).