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## Main Question or Discussion Point

how do you prove that if v is an element of V (a vector space), and if r is a scalar and if rv = 0, then either r = 0 or v = 0... it seems obvious, but i have no idea how to prove it...

- Thread starter broegger
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- #1

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how do you prove that if v is an element of V (a vector space), and if r is a scalar and if rv = 0, then either r = 0 or v = 0... it seems obvious, but i have no idea how to prove it...

- #2

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[tex]ra = 0[/tex]

[tex]rb = 0[/tex]

[tex]rc = 0[/tex]

The solution is that either r equals 0, or a, b, and c all equal 0. And if a, b, and c are 0 then the vector

Is this proof satisfactory? There are probably a lot of ways to prove this.

- #3

matt grime

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Here's the basis free proof. Suppose rv=0, then if r is zero we are done, if not multiply rv=0 by 1/r and we see v=0.

- #4

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Yes... if you pick your basis at (A, B, C) then the vectormatt grime said:That proof requires you to pick a basis. If I pick a different basis, do you know that it still holds?

[tex]r(a - A) = A[/tex]

[tex]r(b - B) = B[/tex]

[tex]r(c - C) = C[/tex]

For this to be true for r <> 0, a must equal A, b must equal B and c must qual C, and thus the vector

- #5

matt grime

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No, that isn't how one does a change of basis (of a vector space: the origin isn't fixed.)

- #6

matt grime

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