Answer
False.
Work Step by Step
False, e.g. if B consists of the vector $(1)$ and C consists of the vector $(2)$, then $[v]_b=[w]_C$ if $v=(2)$ and $w=(1)$ but $v\ne w$.
Differential Equations and Linear Algebra (4th Edition)
by Goode, Stephen W.; Annin, Scott A.
Published by
Pearson
ISBN 10:
0-32196-467-5
ISBN 13:
978-0-32196-467-0
Chapter 4 - Vector Spaces - 4.7 Change of Basis - True-False Review - Page 317: g
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