f (Projection witli an orthonormal basis) In class we learned that if V Ç Rw is a subspace with basis ,a* and A € Rnxk is the matrix with columns aA,…, then projection onto V is achieved by the linear transformation with matrix A(ATA)~lAJ. In this exercise we are going to see how this formula simplifies if we had started with an orthonormal basis of V.
a. Suppose qi,… ,q& is an orthonormal basis of V and Q € Rnxfc is the matrix with columns qi,…, q*-
i. Show that the projection matrix P = Q(QJQ)~lQT is qiq{ + q2Q2 + – + qtqfc-
ii. Using (i) compute proj^b, the projection of b e Rn onto V. (Your answer should be a linear combination of qi,…, q*.)
iii. From (ii), what are the coordinates of projvb in the basis qi,… , q/,?
iv. Use your knowledge of orthogonal projectors to write down the matrix that projects onto V1.
v. Using this projector to find proj^jb.
a. Suppose we find additional vectors so that qA,…, q*., q^+i,…,qn is an orthonormal basis of Rn. Check for yomself that {q/.+i,… ,q„} is an orthonormal basis of Vi.
i. Apply what you learned in (a) to the basis {qA+i,… ,q7,} of V1 to compute proj^ib, the projection of b onto V1.
ii. E(juating your answer above and the answer in (a) (v), express b as a linear combination of qA,…, q„.
iii. What, are the coordinates of b in the basis qJ?… ,qn?
a. (4.1, #17) Let L be the line spanned by (1,1,1)T.
i. Find a vector u so that projection onto L is x^ uuTx.
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ii. Compute the projection of b = I 3 I onto L and L[. Show all work.