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jwqwerty
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1. Homework Statement
Prove that there is an additive identity 0∈R^n: For all v∈R^n, v+0=v2. Homework Equations
Axiom of Real Numbers:
There is an additive identity 0∈R : For all a∈R, a+0=a and o+a=a
3. The Attempt at a Solution
Solution 1 (My own attempt)
: Let v=(v1, v2, v3... vn). Then by axiom of R (as stated above), for every vi (i=1, 2..., n) there exists 0∈R such that vi+0=vi. Thus, there exists 0=(0,0,...0)∈R^n.
Solution 2 (Professor's Attempt)
: Let 0=(0,0,...0)∈R^n
then v+0=(v1, v2 ... vn) + (0,0...0)
=(v1+0, v2+0, ... vn+0)
=(v1, v2 ... vn) - by axiom of R (as stated above)
=v
However, I think solution 2 has a problem that it assumed existence of 0 when the question is asking to prove its existence. Or is my thought wrong? And also, is there anything wrong with solution 1?
Prove that there is an additive identity 0∈R^n: For all v∈R^n, v+0=v2. Homework Equations
Axiom of Real Numbers:
There is an additive identity 0∈R : For all a∈R, a+0=a and o+a=a
3. The Attempt at a Solution
Solution 1 (My own attempt)
: Let v=(v1, v2, v3... vn). Then by axiom of R (as stated above), for every vi (i=1, 2..., n) there exists 0∈R such that vi+0=vi. Thus, there exists 0=(0,0,...0)∈R^n.
Solution 2 (Professor's Attempt)
: Let 0=(0,0,...0)∈R^n
then v+0=(v1, v2 ... vn) + (0,0...0)
=(v1+0, v2+0, ... vn+0)
=(v1, v2 ... vn) - by axiom of R (as stated above)
=v
However, I think solution 2 has a problem that it assumed existence of 0 when the question is asking to prove its existence. Or is my thought wrong? And also, is there anything wrong with solution 1?