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#1 2006-09-19 19:07:32

ryos
Member
Registered: 2005-08-04
Posts: 394

Linear Algebra: Check My Proof

Will you please check this proof? It feels like I'm cheating.

Prove that two vectors, a and b are linearly dependent if and only if a is a scalar multiple of b.

Proof:
If a = kb, then a - kb = 0, and the system is linearly dependent.

If, however, a ≠ kb, then a - kb ≠ 0, and the system is not linearly dependent.

My book gives the hint that we should consider separately the case where a = 0 (the zero vector), but that just seems superfluous and unnecessary to me.

What do you guys think?


El que pega primero pega dos veces.

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#2 2006-09-19 21:33:38

Dross
Member
Registered: 2006-08-24
Posts: 325

Re: Linear Algebra: Check My Proof

What definition of linearly dependant have you been given? As far as I can remember, a and b are linearly dependant iff there exists scalars a and b such that aa+bb = 0.

The first part of your proof correctly shows that a being a scalar multiple of b implies that such scalars exist, and so a and b are linearly dependant. The second part of your proof, I think, can be made more rigorous by saying that a is not a scalar multiple of b implies that there exists no k such that a=kb, so the scalars required for linear dependance do not exist.

All in all though, I think your proof is valid.


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#3 2006-09-21 14:36:05

ryos
Member
Registered: 2005-08-04
Posts: 394

Re: Linear Algebra: Check My Proof

Thanks; I got full credit.


El que pega primero pega dos veces.

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