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Prove an if and only if condition for two lines to intersect in Rn

1. Let and be two lines in . Prove that they intersect if and only if is in the linear span of and .
2. Consider the two lines in , Determine whether they intersect.

1. Proof. Assume and intersect. Since and we know there exists an such that . This implies This implies is in the linear span of .

Conversely, assume is in the linear span of . Then there exist such that Thus, there is some point in both and so they intersect 2. The two given lines do not intersect. We know from part (a) that two lines and intersect if and only if is in the linear span of . In this case we have and , . For to be in the linear span of we must have such that But the second equation implies . The third equation would then require which gives . Then, . But then from the first equation Hence, there are no such so is not in the linear span of . Hence, these two lines do not intersect.