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

[gotmejerry]

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!

[HallsofIvy]

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.

[gotmejerry]

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.

[gotmejerry]

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.