Question

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 <b>NOT</b> linearly independent then v3 <b>IS</b> 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.

Answer 1

It is possible that a3 is equal to zero, in which case v3 is not a linear combination of the other three vectors.