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If $\Bbb{C}$ is the field of complex numbers, which vectors in $\Bbb{C}^3$ are linear combinations of $(1, 0, -1)$, $(0, 1, 1)$, and $(1, 1,1)$?

I didn't understand the question actually. It's an exercise in Hoffman's linear algebra book (page 41, exercise 3).

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    @JenniferDylan: oh, okay. I still think this is an incomplete question.2012-08-31

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Well, you can make $(1, 0, 0) = (1, 1, 1) - (0, 1, 1)$.

You can make $(0, 0, 1) = (1, 1, 1) - (0, 1, 1) - (1, 0, -1)$

Therefore, we can get $(0, 1, 0) = (0, 1, 1) - (0, 0, 1) = 2 * (0, 1, 1) - (1, 1, 1) + (1, 0, -1)$.

So, we can write $(1, 0, 0), (0, 1, 0), (0, 0, 1)$ as linear combinations of these. What does that tell us?

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    @TalBin Yes, this means that for any vector, $(x, y, z)$ in $\mathbb{C}^3$, there exist $c_1, c_2, c_3$ such that $c_1 (1, 0, 1) + c_2 (0, 1, 1) + c_3 (1, 1, 1) = (x, y, z)$. And, to show this, instead of figuring out $c_1, c_2, c_3$ directly for every possible vector, I showed that you can make the standard basis vectors, $(1, 0, 0), (0, 1, 0), (0, 0, 1)$, because then it is obvious that every other vector can be formed as a linear combination of these.2012-09-04
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You need to figure which vectors $(a,b,c)$ can be written as

$c_1(1,0,1)+c_2(0,1,1)+c_3(-1,1,1)=(a,b,c)$

This is s system of equations in $c_1,c_2,c_3$, and the question asks you for which $(a,b,c)$ does this system have solution.

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    @TalBin: this is precisely the definition of a linear combination. If someone were to give you a vector $(a,b,c)$ you could try to determine what the values of $c_1, c_2, c_3$ are. When you do, you will find out what vectors it works for. It is a set of simultaneous equations that has been made fairly easy to solve.2012-08-31
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HINT: $\det\pmatrix{1 &0 &1\\0&1&1\\-1&1&1}\neq 0$

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    Why not starting to learn something about them: *...the determinant of a matrix (with real or complex entries, say) is zero if and only if the column vectors of the matrix are linearly dependent.* ([Determinant/Applications/Linear Dependance](http://en.wikipedia.org/wiki/Determinant#Linear_independence))2012-08-31