Is there a difference between the linear independence of ##\{x,e^x\}## and ##\{ex,e^x\}##? It can be shown that both only have the trivial solution when represented as a linear combination equal to zero. However, the definition of linear independence is: "Two functions are linearly independent on the interval ##I## if there exists only the trivial solution to ##c_1f_1 + c_2f_2 + ... + c_nf_n = 0## for all x in ##I##. In the first case, this is obvious since x and e^x never intersect, and so cannot be multiples of each other. However, doesn't the latter case violate this definition since ##e(1)## is a multiple of ##e^1##? I am just confused about the "for all x on I" statement at the end of the definition.(adsbygoogle = window.adsbygoogle || []).push({});

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# I Linear independence of functions

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