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Please anwser in detail Determine if the following sets of vector are linearly independent. If not,...

Please anwser in detail

Determine if the following sets of vector are linearly independent. If not, write one vector as a linear combination of other vectors in the set.

[1 2 -4] , [3 3 2] , [4 5 -6]

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Answer #1

Given set of vectors is = {(1,2,-4),(3,3,2),(4,5,-6)}.

Let P = (1,2,-4), Q = (3,3,2), R = (4,5,-6).

Let us consider the relation aP + bQ + cR = , where a,b,c are real numbers.

Then a(1,2,-4) + b(3,3,2) + c(4,5,-6) = (0,0,0).

Therefore, a + 3b + 4c = 0....(i), 2a + 3b + 5c = 0....(ii), -4a + 2b - 6c = 0.....(iii)

Subtracting (i) from (ii) we get, a + c = 0, i.e., a = -c

Putting this in (iii) we get, 4c + 2b - 6c = 0, i.e., 2b - 2c = 0, i.e., b = c

Putting a = -c and b = c in (i) we get, -c + 3c + 4c = 0, i.e., 6c = 0, i.e., c = 0

Therefore, a = 0, b = 0, c = 0.

Hence, the given set of vectors is linearly independent.

And, none of the vectors can be expressed as the linear combination of other vectors in the set.

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