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Apply the Gram–Schmidt algorithm to these vectors to obtain an orthonormal basis f1, f2, f3.

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

Problem 1.   

Apply the Gram–Schmidt algorithm to these vectors to obtain an orthonormal basis f1, f2, f3.

Problem 2 and 3. Consider the two dimensional  vector space V  = span(v1, v2), where v1, v2 are given in Problem 1 above. Consider the projection of vectors in R4 onto this space V . Use two different methods to find the matrix P of this projection.

Problem 2.

Method I) Follow the procedure described in Question 2 (3) of the in-class questions on Oct 20 to find the projection matrix P .

Link: https://rutgers.instructure.com/courses/74179/files/12323143/download?wrap=1

Problem 3.

Method II) Utilize an orthonormal basis to find the projection matrix P .

In Problem 1 above, you’ve found an orhthonormal basis f1, f2 for the space V = span(v1, v2) = span(f1, f2).  Given a generic y R4, let yJ = Py denote the projected vector. Express yJ

as a linear combination as

Py = yJ = b1f1 + b2f2 (*)

(1) Use the fact that f1, f2 are orthonormal to find b1 b2 in terms of f1, f2, and y. Instruction: operator with symbols f1, f2, y. Do not insert the values of f1, f2 yet.

(2) Insert your answer in (1) back into (*). Identity the matrix P from your result. Express your answer for P in terms of f1, f2.

Instruction: operator with symbols f1, f2.  Do not insert the values of f1, f2 yet.

(3) Now insert in the numerical values of f1, f2  (from Problem 1 above) into your answer for P in (2), and check that the result agrees with what you found in Problem 2.

Problem 4. Given a vector space V in Rn. Consider the projection of vectors in Rn onto V , and let P denote the projection matrix. Follow the steps below to prove that P 2 = P and P T = P .

(1) Let f1, . . . , fk  be an orthonormal basis for  V . (Such an orthonormal basis  always  exists. We won’t worry about proving its existence here.) Express the projection matrix P  in term of f1, . . . , fk.

Hint: generalize the procedure in Problem 3 above.

Instruction: explain all your derivation. Simply showing the answer does not account for a full solution.

(2) Use your answer in (1) to verify P 2 = P and P T = P .

Remark: In Problem 4 of Homework 6, you’ve shown that if P 2 = P and P T  = P  then P  is a projection matrix. Problem 3 here gives the converse statement. Combining these two statements gives

“A square matrix P  is a projection matrix if and only if P 2 = P and P T = P .”

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