Chapter 4 - Vector Spaces - 4.5 Linear Dependence and Linear Independence - Problems - Page 297: 41

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Given: $f_1(x)=x$ $f_2(x)=x$ if $x\ne0$ $f_2(x)=1$ if $x =0$ Let $a,b$ be scalars such that $af_1(x)+bf_2(x)=0 \forall x \in (-\infty, \infty)$ If we assume $x=1$ then $a+b=0$ if $x=0 \rightarrow b=0$ Hence, the linear combination of $f_1$ and $f_2$ is trivial and $f_1,f_2$ are also linearly independent on $(-\infty, \infty)$