I have the following set: $\{ [1; -1; -2], [-1;0;1], [1;2;1] \}$ and I need to find out whether the set is independent or dependent. My answer and the book's answer contradict. I thought it was linearly independent but the book says it is linearly dependent.
For it to be linearly dependent, there has to be a constant scalar that is not $0$ otherwise it wouldn't really prove anything since making all constant scalars $0$ would be trivial. I've tried to find such constants where at least one vector is not multiplied by $0$ but I cannot find one. Also if we take $[1; -1; -2]$ and $[1;2;1]$ we can see that they are not multiples of one another therefore $[-1;0;1]$ has to be multiplied by a constant scalar other than $0$ and be a multiple of $[1;-1;-2]$ and $[1;2;1]$ for us to have linear dependency. However there is a $0$ in the second row of $[-1;0;1]$ and so for $[-1;0;1]$ to be in the span it must be a multiple of the other two vectors but no constant scalar will give us $-1$ or $2$ when multiplied with $0$.
I hope how I worded that makes sense. Basically I've tried looking at it in terms of $c_1 v_1 + c_2 v_2 + c_3 v_3 = 0$ where at least one constant $c_i$ is not a zero. I've also tried looking at it in terms of $c_2 v_2 + c_3 v_3 = v_1$ and $c_2 v_2 + c_1 v_1 = v_3$ where $v_1$ is a linear combination of $v_2$ and $v_3$ or $v_3$ is a linear combination of $v_2$ and $v_1$.
It's good to know general methods for doing this kind of problem. It's also good to be alert to the possibility that some special property of an individual problem allows a trick that gives the answer easily.
In your example, calling the vectors $v_1$, $v_2$, $v_3$, and imagining that $c_1v_1+c_2v_2+c_3v_3=0$ for some real numbers $c_1,c_2,c_3$, we note, by looking at the second coordinate, that $c_1=2c_3$. But then we get $$c_1v_1+c_3v_3=2c_3v_1+c_3v_3=c_3(2v_1+v_3)=c_3(3,0,-3)=-3c_3v_2$$ and we have the linear dependence $2c_3v_1+3c_3v_2+c_3v_3=0$, or $$2v_1+3v_2+v_3=0$$