Math 301 Final Exam

Dr. Holmes

December 17, 2007

The final exam begins at 10:30 am. It ends officially at 12:30 pm; ifeveryone in the class agrees to this, it will continue until 12:45 pm.

The exam is open book, one sheet of notebook paper with notes of yourchoice, any calculator. The advice on calculator use is the same as on thehour exams. The matrix functions which you may use, apart from basiccalculations, are reduction of matrices to row echelon form, multiplication ofmatrices, computation of inverses of matrices, computation of determinantsof matrices. Whenever you use your calculator, explain what you did withit. Be sure to read instructions, as you will in some cases have to carry outthese calculations by hand.

Good luck, and have a happy holiday season!!!

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1. Present the vector equation

x1

231

+ x2

211

+ x3

014

=

64−3

as a system of three simultaneous equations in three unknowns and asa matrix equation Ax = b.

Solve this system of equation by matrix methods (row operations),explicitly listing each row operation used.

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2. Consider the linear transformation which reflects the plane throughthe y-axis then stretches the plane vertically by a factor of two. Writethe matrix which represents this transformation. State its eigenvaluesand associated eigenvectors (you should not actually need to do anycalculations to determine the eigenvalues and eigenvectors, though youcan of course do the problem this way).

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3. (a) Set up and carry out a calculation with inverse matrices whichsolves the system of equations

2x− y = 6

3x+ 2y = −5

You can use your calculator. Be sure to state the final solution interms of values of x and y.

(b) Show that the matrix

2 −3 131 1 −1−1 2 −8

is invertible or write a linear combination of its component columnvectors which adds to the zero vector.

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4. Solve the system of equations by Cramer’s rule. The determinants mustbe set up on paper but may be evaluated by calculator.

x− 2y + 3z = 8

−2x+ y + 5z = 12

3x− y + 2z = 10

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5. (a) Prove that the set of all vectors of the form

2yx− y

3x

is a subspace of R3 Your proof must show explicitly that this sethas each defining property of a subspace.

(b) Prove that the set of all vectors of the form

xx3

2x

is not a subspace of R3. Your proof must show explicitly thatthis set fails to have (at least) one of the defining properties of asubspace.

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6. Find bases for the column space and null space of the following matrix.State its rank.

2 3 2 04 6 −1 56 9 1 5

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7. Compute the eigenvalues and eigenvectors of the matrix

[ −16 −930 17

]

Express the matrix in the form PDP−1 (D a diagonal matrix) if theeigenvalues are real, and in the form PCP−1 (C a matrix of the form[a −bb a

]) if the eigenvalues are complex.

Show your work, and indicate the role of an appropriate characteris-tic polynomial. You may not use built in eigenvector/value findingfunctions except to check.

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8. Find an orthogonal basis for the subspace of R4 spanned by

1111

,

1−110

, and

11−1−1

, using the Gram-Schmidt process.

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9. Each month seven percent of Windows XP users switch to WindowsVista. Each month three percent of Windows Vista customers switchback to XP. Set up the stochastic matrix representing this situation. Ofcourse in month 0 all users have Windows XP. Set up and carry out amatrix calculation determining what percentage of users have WindowsXP after two weeks. Determine the steady state vector (showing allwork): if Microsoft continued to service Windows XP indefinitely whatpercentage of users would stay with Windows XP after a long time? (ofcourse all computer companies and software packages in this problemare completely fictional and bear no resemblance to any real computercompanies or software packages).

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10. Basis B for R2 is

{[13

],

[ −12

]}

and basis C is

{[11

],

[ −14

]}.

Compute the matrix which converts B coordinates to C coordinates(showing all work, of course).

Write down B-coordinates and C-coordinates for the vector with stan-

dard coordinates

[12

]and show a calculation verifying that the matrix

found in the previous part actually converts the B-coordinates of thisparticular vector to its C-coordinates.

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