Science:Math Exam Resources/Courses/MATH152/April 2012/Question 06 (a)
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Question 06 (a) 

This question requires little or no calculations. For what value(s) of a are the following vectors , , , linearly dependent? 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

A set of three vectors is linearly independent if is only satisfied if . 
Hint 2 

For an alternative solution: What is true about the columns of a matrix when its determinant is equal to zero? 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. A set of vectors is linearly dependent if has more solutions than just the trivial solution . To determine which values of will cause the vectors to be linearly dependent, we write the equation in its matrixvector form If the rowreduced echelon form of has a row of zeros, there are nonzero values of that will satisfy . Rowreducing as follows: we can see that the last row will be a row of zeros if . Therefore, the vectors are linearly dependent if . 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. As a quick alternative to the first solution, recall that the columns of a matrix, , are linearly dependent if the determinant of is equal to zero. Hence, we define which has the determinant The determinant equal to zero when . Therefore, the vectors are linearly dependent if . 