Hello everyone. I was going through my Linear Algebra Done Right textbook that threw me off. I hope this forum is appropriate for my inquiry; while there is no problem I'm trying to solve here, I don't know whether just asking for clarification would belong to the homework forum instead. If this is the case, I apologize.(adsbygoogle = window.adsbygoogle || []).push({});

I'll start off by quoting the passage:

"For another example of a linearly independent list, fix a non-negative integer m. Then (1,z,...,z^{m}) is linearly independent in P(F). To verify this, suppose that a_{0}, a_{1},...,a_{m}, belonging to F are such that:

a_{0}+ a_{1}z + ... + a_{m}z^{m}= 0, for every z belonging in F.

If at least one of the coefficients a_{0}, a_{1},...,a_{m}were nonzero, then the above equation could be satisfied by at most m distinct values of z; this contradiction shows that all the coefficients in the above equation equal 0. Hence (1,z,...,z^{m}) is linearly independent, as claimed."

Linear independence, as I understand it, holds only when each vector in a list of vectors has a unique representation as a linear combination of other vectors within that list. It is my interpretation that Axler is specifically using the fact that the {0} vector, in the above polynomial vector space example, can only be expressed by setting all coefficients to 0.

My confusion, I think, stems from how he concludes that all the coefficients must be zero. If any coefficient is nonzero, then the equation has, at most, m roots (I hope I am correctly relating this to the Fundamental Theorem of Algebra). But then, as I see it, this shows that the equation has more than one representation for {0} and is thus not linearly independent. But instead, he uses this same fact to obtain a contradiction and conclude that all the coefficients must equal 0.

Unfortunately, for some reason, the TeX editor did not work properly for me, so I had to resort to expressing some things here differently. Anyway, if anyone could shed some light and point me in the right direction, I would greatly appreciate it.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Linear Independence: Polynomial Example

**Physics Forums | Science Articles, Homework Help, Discussion**