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Locrianz

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He then goed onto say that if a basis is chosen in the subspace L, say f1,f2,...,fr, then additional vectors can be chosen from the linear space K, say f(r+1),...,fn such that f1,f2,...,fr,...,fn form a basis for K. Which makes perfect sense. But here is the part that throws me:

Suppose we have the relation:

c1f1+c2f2+...+crfr+c(r+1)f(r+1)=0

Where the c terms are constants and f(r+1) is an additional vector from K added to the basis of L.

He says if c(r+1) does not equal zero then then L is K. Which is true. But then he says if c(r+1)=0 then the vectors f1,f2,...,fr are linearly dependent, which would be a contradiction. How can this be true? If this is the basis for L and we set it equal to zero then we have all the constants equal to zero, hence they will be linearly independent. I checked the errata and there was no mention of this.

Any help will be appreciated, thanks guys.