anonymous
  • anonymous
proof for if two vectors ar linearly dependent that one is a scalar multiple of the other
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Suppose two vectors u and v are linearly dependent. If either u or v is the zero vector, say u = 0, then u = 0*v. If neither u nor v are zero vectors, then there exists scalars a1 and a2, not both zero, such that a1*u + a2*v = 0. If a1 is 0, then a2 * v = 0, implying a2 = 0 (since v is nonzero). a1 = a2 = 0 contradicts our assumption that we don’t have both scalars being zero. Therefore, a1 can’t be zero. Similarly, a2 can’t be zero. And so they’re both nonzero scalars. Then from a1*u + a2*v = 0, solving for u, we get u = -(a2/a1) * v. So u is a scalar multiple of v.

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