Chapter 4.7, Problem 15E

Exercises 15 and 16 provide a proof of Theorem 15. Fill in a justification for each step.

15. Given v in V. there exist scalars x_{1, ….,} x_n. such that

v = x₁b₁ + x₂b₂ + … + x_nb_n

because (a) ______. Apply the coordinate mapping deter-mined by the basis C. and obtain

[v]_C = x₁[b₁]_C + x₂[b₂]_C + … + x_n[b_n]_C

because (b) ______. This     equation     may be     written     in the form

[v]_C = [[b₁]_C [b₂]_{C …} [b_n]_C] $\left[\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right]$    (8)

by the definition of (c) ______.This shows that the matrix $C\stackrel{P}{\leftarrow }B$ shown in (5) satisfies [v]_C = $C\stackrel{P}{\leftarrow }B$[v]_B for each     v in V, because the vector on the right side of (8) is (d) ______.