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