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kingwinner
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Homework Statement
Claim:
The solution space of a linear homogeneous PDE Lu=0 (where L is a linear operator) forms a "vector space".
Proof:
Assume Lu=0 and Lv=0 (i.e. have two solutions)
(i) By linearity, L(u+v)=Lu+Lv=0
(ii) By linearity, L(au)=a(Lu)=(a)(0)=0
=> any linear combination of the solutions of a linear homoegenous PDE solves the PDE
=> it forms a vector space
Homework Equations
N/A
3. The Attempt at a Solution and comments
Now, I don't understand why ONLY by proving (i) and (ii) alone would lead us to conclude that it is a vector space. There are like TEN properties that we have to prove before we can say that it is a vector sapce, am I not right?
Are there any theorem or alternative definition that they have been using?
Thanks!
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