Linearly Independent Proof

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If your going to uses acronyms please specify them first. It takes a little effort and it means people can understand you! Your post is pure Gobbledegook! You'll find it on Google!

Please reformulate this in a browser with a toolbar that allows proper formatting of subscripts and symbols such as ≤.

Engineering and pure math are not the same; I would bet that less than 0.1% of CR4 members would know what a "normed linear space" is; e.g., I've long since forgotten.

Use δ and ε rather than "delta" and "e".

WLOG = "without loss of generality"; RHS = ??

Part of the problem states that:

c1y1+...+ckyk=0

and

WLOG suppose c1=1

At the end you say: I know i need to use that fact that the norms are equivalent to prove that c1 cannot equal 1, but I don't know how to do it.

I'm not sure that this statement is true (that c1 ≠ 1). Consider the equivalent equation:

c1'y1+...+ck'yk=0, where c1' ≠ 0, and ≠ 1. Divide both sides of the equation by c1' and you get:

c1'y1/c1' + c2'y2/c1' + ... ck'yk/c1' = 0, which can be rewritten as:

c1y1 + c2y2 + ... ckyk = 0; where c1 clearly equals 1 ( and c2'/c1' = c2, etc).

Thus c2 can equal 1.

The final statement should, of course, have been:

Thus c1 can equal 1.