Math Is Fun Forum

deepz

Member

Registered: 2007-03-20

Posts: 10

linear algebra

Let u,v and w be vectors in a vector space. Prove that if u+v=w+v then u=w.

I think its something to do using the cancellation law along with the zero element and inverse axioms of vector spaces. Can someone help me clarify this, thanks!

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: linear algebra

Add the vector −v to both sides, and then use the associative law. This is how you prove the cancellation law for groups in general. Remember that vectors in a vector space form an Abelian group under vector addition.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: linear algebra

one of the rules for a vector space is that for any element v of a vectorspace V there must be an ellement -v in V such that v + (-v) = 0 or the Zero element (we make a distinction between the zero element and the number zero)

so yeah, as jane said, if you have u+v = w + v, then clearly u+v + (-v) = v + w + (-v) so u + 0 = w + 0 so u = w.

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