For which L(s) will be these vectors linearly dependent?

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The discussion centers on determining the values of L for which the vectors a = [1, 1, 1], b = [2, L, 0], and c = [L, 2, 3] are linearly dependent. The participants conclude that L can take the values 1 or 2, which results in the vectors being dependent. Additionally, they explore the implications of L on the vector v = [ß, 0, -1] being in the span of a, b, and c, noting that if L is restricted to these values, then ß can be any real number.

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gotmejerry
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So i have 3 vectors:
a= [1 1 1]
b= [2 L 0]
c= [L 2 3]

How do I calculate the L in order to make these vecotrs linearly dependent?

How does ß depend from L if v= [ß 0 -1] and v is in span(a b c)?

Thank you!
 
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Do you know what "linearly dependent" means? Use the definition of "linearly dependent" You will get a cubic equation for L but, fortunately, there's 1 obvious root.

For the second question, are we to assume that L is one of those values? Otherwise, the span of a, b, and c is all of R3 and \beta can be anything.
 
I guess I know.

I wrote 3 equations:

α + 2β + Lγ = 0
α + Lβ + 2γ = 0
α + + 3γ = 0

And i got, L can be 1 or 2. Then I checked it and for these Ls the vectors are dependents. But how do I know that there aren't more Ls.

For the second question. Yes L is from those values.
 
HallsofIvy said:
Do you know what "linearly dependent" means? Use the definition of "linearly dependent" You will get a cubic equation for L but, fortunately, there's 1 obvious root.

For the second question, are we to assume that L is one of those values? Otherwise, the span of a, b, and c is all of R3 and \beta can be anything.

But I maybe got the definition wrong.
 

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