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.
It is possible that a3 is equal to zero, in which case v3 is not a linear combination of the other three vectors.
Therefore, if v3 is not a linear combination of v1, v2, and v4, it does not imply that {v1, v2, v3, v4} is linearly dependent. In fact, it is possible for {v1, v2, v3, v4} to be linearly independent even when v3 is not a linear combination of the other vectors. Hence, the statement is false.
To show that {v1, v2, v3, v4} is linearly dependent, we need to find a nontrivial solution to the equation a1 v1 + a2 v2 + a3 v3 + a4 v4 = 0, where not all of the coefficients a1, a2, a3, a4 are zero.
If v3 is not a linear combination of v1, v2, v4, then a3 must be nonzero for this equation to hold. However, this doesn't necessarily mean that v3 is a linear combination of v1, v2, v4.
For example, consider the vectors v1 = [1, 0, 0, 0], v2 = [0, 1, 0, 0], v3 = [0, 0, 1, 0], and v4 = [0, 1, 1, 1]. If we choose a1 = 1, a2 = 1, a3 = 1, and a4 = -1, we have:
1 * [1, 0, 0, 0] + 1 * [0, 1, 0, 0] + 1 * [0, 0, 1, 0] + (-1) * [0, 1, 1, 1] = [0, 0, 0, 0]
Here, not all of the coefficients are zero, and the equation is satisfied. However, v3 = [0, 0, 1, 0] cannot be expressed as a linear combination of v1, v2, v4.
Therefore, the statement is false. The fact that v3 is not a linear combination of v1, v2, v4 does not imply that {v1, v2, v3, v4} is linearly independent.