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Homework Statement
When is |x+y|=|x|+|y| for arbitrary non-zero vectors x,y∈Rn ie, when does equality hold for the well known inequality |x+y|≤|x|+|y|
Homework Equations
|x|2=<x,x>=Σixi2
|x+y|≤|x|+|y|
The Attempt at a Solution
squaring both sides of the inequality we have
(|x+y|)2≤|x|2+|y|2+2|x||y|,
but (|x+y|)2=∑i(xi+yi)2=∑ixi2+∑iyi2+2∑ixiyi, then, by definition of norm we can cancel out all terms in the inequality but
2∑ixiyi=<x,y>≤2|x||y| so we see that equality can only hold when both sides of this inequality are equal, therefore we square both sides
<x,y>2=|x|2|y|2, by definition this expands to ∑ixiyi⋅∑ixiyi=
∑i(xiyi)2+∑i≠jxixjyiyj
but ∑i(xiyi)2=|x|2|y|2
so our condition for equality of |x+y|=|x|+|y| simplifies to ∑i≠jxixjyiyj=0 or
∑i≠jxixjyiyj=0
this is the answer I arrived at, however I don't feel confident in it for some reason. I found other solutions online for the problem in this exact book that state "when x and y are linearly dependent", however in the newest edition of the book, which I have, the problem actually states : "hint: the answer is not 'they are linearly dependent.'"
Edit: I've already found some errors in my work, including the fact that I did not expand (|x||y|)^2 properly
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