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Homework Statement
Is {(1,4,-6), (1,5,8), (2,1,1), (0,1,0)} a linearly independent subset of R^3. Justify your answer
Homework Equations
The Attempt at a Solution
I asssumeda(1,4,-6) + b(1,5,8) + c(2,1,1) + d(0,1,0) = 0
then i set up the system
a + b + 2c = 0
4a + 5b + c + d = 0
-6a + 8b + c = 0
My first step was to switch the 2nd row with the 3rd row:
a + b + 2c = 0
-6a + 8b + c = 0
4a + 5b + c + d = 0
then i replaced the second row with ( 6R1 + R2)
and replaced the third row with (-4R1 + R3)
my result is
a+ b + 2c = 0
14b + 13c = 0
b - 7c + d = 0
then i replaced thethird row with ( - 1/14 R2 + R3)
a + b + 2c = 0
14b + 13c = 0
-111/14 c + d = 0
on this step, it's looking closer to what Dick got, and is there supposed to be another manipulation with the rows?
I just solved for d = 111/14 c
then i just let c = 1, thus
c = 1
d = 111/14
b = -13/14
a = (13/14) - 2
but if i let c = 14
i get dick's answer
c = 14
d = 111
b = -13
a = -15are both results correct??
and if not, (meaning Dick's is the only correct solution), what is the next step in the algorithm to find c = 14?
thanks for the helpnevermind, upon further reading i found that
"In this case, the system does not have a unique solution, as it contains at least one free variable. The solution set can then be expressed parametrically (that is, in terms of the free variables, so that if values for the free variables are chosen, a solution will be generated)."
so there's no unique solution, since you can choose whatever you want your variable "c" to be.
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