If v1,...,v4 are in R^4 and v3 is not a linear combination of v1, v2, v4 then {v1, v2, v3, v4] is linearly independent.
Is this true or false? Why?
If v1,...,v4 are in R^4 and v3 is not a linear combination of v1, v2, v4 then {v1, v2, v3, v4] is linearly independent.
This is equivalent to the statement:
If {v1, v2, v3, v4} is NOT linearly independent then v3 IS a linear combination of v1, v2, v4 .
This is false. If {v1, v2, v3, v4} is linearly dependent, then that means that there exists real numbers a1, a2, a3 and a4, not all equal to zero such that:
a1 v1 + a2 v2 + a3 v3 + a4 v4 = 0
Now, if a3 is not zero, then you can write v3 as a linear combination of the other three vectors, but this need not be the case.
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It is possible that a3 is equal to zero, in which case v3 is not a linear combination of the other three vectors.
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