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MATH 304 Test 3 - Matrix Algebra and Linear Transformations, Exams of Linear Algebra

Western Washington University (WWU)Linear Algebra

The march 1, 2005 test for math 304, a university-level course on matrix algebra and linear transformations. The test covers topics such as eigenvectors, unit vectors, singular value decomposition, subspaces, and quadratic forms. Students are required to solve problems related to finding eigenvectors and eigenvalues, orthogonal matrices, singular value decompositions, bases for orthogonal complements, and maximizing quadratic forms subject to certain constraints.

Typology: Exams

Pre 2010

Uploaded on 08/16/2009

koofers-user-ot9
koofers-user-ot9 🇺🇸

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MATH 304 Test 3
March 1, 2005 Name
Put your answers in the space provided. Show your reasoning. The maximum score on the test is 30 points.
Calculators may be used unless specifically restricted.
1. 3 points Let u= [1/3,−2/3,2/3]T. Show that uis an eigenvector for uuTand find all of its eigenvalues.
Do not use your calculator.
Answer
2. 4 points Let ube a unit vector in Rn. Let Q=I−2uuT. Show that Qwill always be an orthogonal
matrix.
pf3
pf4

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Download MATH 304 Test 3 - Matrix Algebra and Linear Transformations and more Exams Linear Algebra in PDF only on Docsity!

MATH 304

Test 3

March 1, 2005 Name

Put your answers in the space provided. Show your reasoning. The maximum score on the test is 30 points.

Calculators may be used unless specifically restricted.

  1. 3 points Let u = [1/ 3 , − 2 / 3 , 2 /3]T^. Show that u is an eigenvector for uuT^ and find all of its eigenvalues.

Do not use your calculator.

Answer

  1. 4 points Let u be a unit vector in Rn. Let Q = I − 2 uuT^. Show that Q will always be an orthogonal

matrix.

  1. 5 points Find a singular value decomposition, SVD, for the matrix A =

 

 . Circle your answer.

  1. Consider the quadratic form Q(x) = 2x^21 + 3x^22 + 2x^23 + 2x 1 x 2 + 4x 1 x 3 + 2x 2 x 3.

6a. 3 points Find the maximum value of Q(x) subject to the constraint ‖x‖ = 1.

Answer

6b. 3 points Find a unit vector u where this maximum is attained.

Answer

  1. 4 points Find P and the new quadratic form when one makes the change of variable, x = P y that

transforms the quadratic form Q(x) = x^21 − x^22 + 4x 1 x 3 − 4 x 2 x 3 into a quadratic form with no cross-product

term. Circle your answer.