Science:Math Exam Resources/Courses/MATH307/April 2005/Question 03 (b)
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Question 03 (b) 

Let P be a real symmetric n x n matrix such that P^{2} = P. Also let R = I_{n}  2P be the matrix that reflects a vector across a plane. (b) If P is the matrix of the projection of onto a subspace V and if Q is the matrix of the projection of onto the orthogonal complement , explain (either algebraically or geometrically) why PQ = 0 and RQ = Q. 
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Hint 

Science:Math Exam Resources/Courses/MATH307/April 2005/Question 03 (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. Why is is in the subspace and will be in the subspace orthogonal to . So it is easy to imagine that if is in then will be the zero vector meaning . To get a vector in from any vector in we simply use . Substituting this into the first equation we get and because x does not have to be a zero vector . The same logic is true for the opposite case so . To see RQ = Q we can solve algebraically
but as seen above so 