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# Prove some facts about a given set of vectors

1. Prove that the vectors are linearly independent in .

2. Prove that the vectors are linearly dependent in .

3. Find all such that the vectors are linearly dependent in .

1. Proof. We consider the equation From the first equation we have and from the third we have . Plugging these into the second we have which implies . Hence, , so the vectors are independent 2. Proof. Consider the equation The first and third equations implies . By the second equation we then have which implies is arbitrary (since this will hold for all choices of ). Letting , we have and we have the equation Hence, these vectors are linearly dependent 3. For , , to be independent we must have for some nontrivial choice of . This gives us the equations If then and can be any value, so the vectors are dependent.
Assume . From the first equation we have , and from the third we have . Therefore, the second equation becomes Thus, we have .
Therefore, the vectors are dependent if .