Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 4 - Vector Spaces - 4.4 Spanning Sets - Problems - Page 283: 43

Answer

See below

Work Step by Step

Given $V=C^\infty(I)$ and $S$ be the subspace of $V$ a) For any vector $r(x) \in span \{f(x),g(x)\}$ we have $r(x)=af(x)+bf(x)\\ =a\frac{e^x+e^{-x}}{2}+b\frac{e^x-e^{-x}}{2}\\ =\frac{ae^x+ae^{-x}}{2}+\frac{be^x-be^{-x}}{2}\\ =\frac{a+b}{2}e^x+\frac{a-b}{2}e^{-x}$ b) From exercise a) we can notice that any vector $r(x) \in$ span $\{f(x),g(x)\}$ can be written as a linear combination of $h(x)=e^{-x}$ and $j(x)=e^{-x}$ Hence, $S$ is also spanned by $\{h(x), j(x)\}$
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