Consider the vectors v1 = (1, 0, 2) and v2 = (0, 1, -1). We want to know whether they are linearly independent (ie none is a combination of the other) or dependent (ie one can be expressed as a combination of the other).
We set up the equation for a linear combination:
C1 * V1 + C2 * V2 = 0
Where c1 and c2 are unknown coefficients. Expanding this equation, we get a system of three equations:
C1 = 0
C2 = 0
2C1 – C2 = 0
The only solution to this system is c1 = c2 = 0, which means the only way to get a zero vector is for both coefficients to be zero.
Therefore, v1 and v2 are linearly independent! In general, for n vectors in an n-dimensional space, if the only solution to the same equation with all coefficients equal to zero is the trivial solution, then the vectors are independent.
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