## Exponential Functions Form a Basis of a Vector Space

## Problem 590

Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let

\[V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}\]
be a subset in $C[-1, 1]$.

**(a)** Prove that $V$ is a subspace of $C[-1, 1]$.

**(b)** Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$.

**(c)** Prove that

\[B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}\]
is a basis for $V$.