Science:Math Exam Resources/Courses/MATH307/December 2012/Question 03 (d)

MATH307 December 2012

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Question 03 (d)
Let $S=\left\{[x_{1},x_{2},x_{3}]^{T}:\,x_{1}+x_{2}+x_{3}=0\right\}$ be the subspace of vectors in $\mathbb {R} ^{3}$ 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.

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 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
The matrix P that projects onto the range of B is given by P = B(BTB)-1BT.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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]

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