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If s = {r, u, d} is a set of linearly dependent vectors. if x = 5r + u + d, determine whether t = {r, u, x} is a linearly dependent set

User NZJames
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Consider any arbitrary linear combination of the vectors
\mathbf r,\mathbf u\,\mathbf x. We have


c_1\mathbf r+c_2\mathbf u+c_3\mathbf x=c_1\mathbf r+c_2\mathbf u+c_3(5\mathbf r+\mathbf u+\mathbf d)

=(c_1+5c_3)\mathbf r+(c_2+c_3)\mathbf u+c_3\mathbf d

=c_4\mathbf r+c_5\mathbf u+c_6\mathbf d

We know
\mathbf r,\mathbf u,\mathbf d are linearly dependent, which means there must exist some choice of not all zero constants
c_4,c_5,c_6 such that the combination above gives the zero vector. So
T=\{\mathbf r,\mathbf u,\mathbf x\} is a set of linearly dependent vectors.
User Aleixfabra
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