Science:Math Exam Resources/Courses/MATH221/April 2013/Question 10
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Question 10 

Let Find a basis for the orthogonal complement of W. 
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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Are all three of the vectors linearly independent? If not, what is the dimension (and basis) of W. What will the dimension of the basis of be? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We first note that there is a nontrivial linear relationship between the three vectors, namely Thus, we get that has dimension 2 and thus is also 2 dimensional. Let be the matrix of basis vectors to , If is a basis vector to then orthogonality tells us that Computing this we get the following relationships Since the dimension (or rank) of V is two we will have two free parameters. In particular, this means that the basis is not unique (it never is). We choose and as free parameters to get and finally that Therefore the two orthogonal vectors are a basis for . Notice that the dimension of is two, which happens to be the same dimension as that of . 