{v1, v2, v3} linearly independent if and only if
span {v1, v2, v3} is 3D
{v1, v2, v3} linearly independent if and only if the only solution to the linear combination:
c1*v1 + c2*v2 + c3*v3 = 0
is the trivial solution, where c1, c2, and c3 are scalars. In other words, the only way to obtain the zero vector as a linear combination of v1, v2, and v3 is by setting all the coefficients to zero.
This statement can be proved using the definition of linear independence, which states that a set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. If {v1, v2, v3} is linearly independent, then no vector in the set can be written as a linear combination of the others, which means that the only way to obtain the zero vector is by setting all the coefficients to zero. On the other hand, if {v1, v2, v3} is not linearly independent, then at least one vector in the set can be written as a linear combination of the others, which means that there exists a nontrivial solution to the linear combination above that results in the zero vector.
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