determine whether the vectors (1,3,-1,4),(3,8,-5,7),(2,9,4,23) in R^4 are linearly independent or linearly dependent
1
Expert's answer
2021-06-15T16:33:47-0400
Let us determine whether the vectors (1,3,−1,4),(3,8,−5,7),(2,9,4,23) in R4 are linearly independent or linearly dependent. Let us consider the equality
a(1,3,−1,4)+b(3,8,−5,7)+c(2,9,4,23)=(0,0,0,0).
It follows that (a+3b+2c,3a+8b+9c,−a−5b+4c,4a+7b+23c)=(0,0,0,0), and hence we have the following system:
⎩⎨⎧a+3b+2c=03a+8b+9c=0−a−5b+4c=04a+7b+23c=0
After adding to the second equation the first multiplied by -3, to the third equation the first, and to the fourth equation the first multiplied by -4, we conclude that the last system is equivalent to following the system:
⎩⎨⎧a+3b+2c=0−b+3c=0−2b+6c=0−5b+15c=0
After dividing the third equation by 2 and the fourth equation by 5, we conclude that the second equation coinside with the third equation and the fourth equation. Therefore, we have the following equivalent systems:
⎩⎨⎧a+3b+2c=0b=3cc∈R
⎩⎨⎧a=−3b−2c=−11cb=3cc∈R
Let c=1, then b=3,a=−11. We have that −11(1,3,−1,4)+3(3,8,−5,7)+(2,9,4,23)=(0,0,0,0), and hence we conclude that this vectors are linearly dependent.
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