Science:Math Exam Resources/Courses/MATH307/December 2012/Question 03 (d)
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Question 03 (d)
be the subspace of vectors in whose components sum to zero.
(d) Write down the MATLAB/Octave code that
(i) computes the projection matrix P that projects onto S and
(ii) computes the vector in S that is closest to [0, 1, 0]T.
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The matrix P that projects onto the range of B is given by P = B(BTB)-1BT.
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To begin with, we define the matrix B using the result from part (c):
B = [-1 -1; 1 0; 0 1];
Then, use the formula for the projection matrix to define P:
P = B*inv(B'*B)*B';
This solves (i). Note that all possible choices of B would have resulted in the same projection matrix P. Finally, the vector x in S closest to [0, 1, 0]T is the projection of that vector onto S, that is:
x = P*[0; 1; 0]