Math 301 Final Exam. Dr. Holmes. December 17, 2007

Transcription

1 Math 30 Final Exam Dr. Holmes December 7, 2007 The final exam begins at 0:30 am. It ends officially at 2:30 pm; if everyone in the class agrees to this, it will continue until 2:45 pm. The exam is open book, one sheet of notebook paper with notes of your choice, any calculator. The advice on calculator use is the same as on the hour exams. The matrix functions which you may use, apart from basic calculations, are reduction of matrices to row echelon form, multiplication of matrices, computation of inverses of matrices, computation of determinants of matrices. Whenever you use your calculator, explain what you did with it. Be sure to read instructions, as you will in some cases have to carry out these calculations by hand. Good luck, and have a happy holiday season!!!

2 . Present the vector equation 2 x 3 + x x = as a system of three simultaneous equations in three unknowns and as a matrix equation Ax = b. Solve this system of equation by matrix methods (row operations), explicitly listing each row operation used. 2

3 2. Consider the linear transformation which reflects the plane through the y-axis then stretches the plane vertically by a factor of two. Write the matrix which represents this transformation. State its eigenvalues and associated eigenvectors (you should not actually need to do any calculations to determine the eigenvalues and eigenvectors, though you can of course do the problem this way). 3

4 3. (a) Set up and carry out a calculation with inverse matrices which solves the system of equations 2x y = 6 3x + 2y = 5 You can use your calculator. Be sure to state the final solution in terms of values of x and y. (b) Show that the matrix is invertible or write a linear combination of its component column vectors which adds to the zero vector. 4

5 4. Solve the system of equations by Cramer s rule. The determinants must be set up on paper but may be evaluated by calculator. x 2y + 3z = 8 2x + y + 5z = 2 3x y + 2z = 0 5

6 5. (a) Prove that the set of all vectors of the form 2y x y 3x is a subspace of R 3 Your proof must show explicitly that this set has each defining property of a subspace. (b) Prove that the set of all vectors of the form x x 3 2x is not a subspace of R 3. Your proof must show explicitly that this set fails to have (at least) one of the defining properties of a subspace. 6

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

8 7. Compute the eigenvalues and eigenvectors of the matrix [ 6 ] Express the matrix in the form P DP (D a diagonal matrix) if the eigenvalues [ ] are real, and in the form P CP (C a matrix of the form a b ) if the eigenvalues are complex. b a Show your work, and indicate the role of an appropriate characteristic polynomial. You may not use built in eigenvector/value finding functions except to check. 8

9 8. Find an orthogonal basis for the subspace of R 4 spanned by, 0, and, using the Gram-Schmidt process. 9

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

11 {[ 0. Basis B for R 2 is 3 and basis C is {[ ], ] [, 2 [ 4 ]}. ]} 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 standard coordinates and show a calculation verifying that the matrix 2 found in the previous part actually converts the B-coordinates of this particular vector to its C-coordinates.