Let u,v be distinct vectors in a vector space V. Show that {u,v} is linearly

dependent if and only if u or v is a multiple of the other.

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- June 2nd 2008, 06:38 PMJCIRLinear independence
Let u,v be distinct vectors in a vector space V. Show that {u,v} is linearly

dependent if and only if u or v is a multiple of the other. - June 2nd 2008, 07:00 PMTheEmptySet
So we have a statement

where p is if u,v are linearly dependent and q is if u or v is a multiple of the other

Since we are proving and if and only if statement we need to prove both directions. So lets start with

We assume that u and v are linearly dependant. So by definition there exits scalers such that and Now if we solve this equation for u we get . Therefore u and v are multiples of each other. Done.

Now for the other direction

We assume that there are mulitples of each other so we get

where Now we subract v from both sides to get

Now we have to non zero scalers and a linear combination that is equal to zero. So the vectors are dependant.

QED.

I hope this helps. - June 2nd 2008, 07:06 PMJCIR
Yes it helps greatly.