Science:Math Exam Resources/Courses/MATH307/April 2006/Question 05 (b)
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Question 05 (b) 

A matrix A and a vector b are given by
Find an orthonormal basis for in which span the column space R(A) of the matrix A. How does the third basis vector relate to the fundamental subspaces of A? 
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Hint 

Science:Math Exam Resources/Courses/MATH307/April 2006/Question 05 (b)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. is orthogonal to the R(A) since it is a part of the orthonormal basis for Recall that Since the dim of , the basis of is We can find either by finding or using gramschmidt process or with cross products We solve it by finding
Normalize the vector
Alternatively, if you did not recognize that we can still compute with cross product
And we note our answer from computing the nullspace versus the cross product is a multiple of 1 but either one can be used as for the orthonormal basis of 