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I have a question in which vector1 ( 0,-2,2) and vector2(1,3,-1) are given and we need to check that are these vectors linear combination of w(0,4,5) or not. After solving this Rank =3 i.e greater than number of unknown variable i.e.2.

Rank > n So is it linear combination or not ?

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  • $\begingroup$ "Resultant" is typically used to describe the sum of two or more vectors. That is, the resultant vector of $(0, -2, 2)$ and $(1, 3, -1)$ is $(1, 1, 1)$. What exactly are you trying to do? Are you trying to determine if $(0, 4, 5)$ is a linear combination of $(0, -2, 2)$ and $(1, 3, -1)$? $\endgroup$ – apnorton May 7 '14 at 2:05
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Short answer: No.

Long answer: No, because when finding the rank, you are essentially whether the $3$ vectors are linearly independent. A matrix of full rank would imply that the vectors ARE linearly independent, hence no linear combination exists between them.

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  • $\begingroup$ Let suppose we have 3 vectors instead of 2 and we get full rank of matrix. It won't be linear combination in that case too ? $\endgroup$ – Wajiha Ali May 7 '14 at 2:22
  • $\begingroup$ @JiyaCIS Sorry for the slow reply. If you include $3$ vectors and your "resultant" vector, so that you have $4$ vectors, you will never have full rank because the rank must be less than or equal to $3$, hence there WILL be a linear combination amongst the $4$ vectors. $\endgroup$ – BlackAdder May 7 '14 at 7:03
  • $\begingroup$ Okay got you thanks. And I have corrected my mistake of "resultant" vector if you see (: $\endgroup$ – Wajiha Ali May 7 '14 at 10:03

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