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Final

MATH240 BOYLE-M FALL2011 0101 FINAL EXAMExam


Department
Mathematics
Course Code
MATH 240
Professor
All
Study Guide
Final

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MATH 240 – Fall 2011 – Final Exam
CALCULATORS ARE NOT ALLOWED
TURN OFF ALL ELECTRONIC DEVICES
*** THERE ARE QUESTIONS ON BOTH SIDES OF THIS PAPER ***
Answer each question on a separate sheet of paper. Use the back side if necessary.
On each sheet, put your name and your section TA and meeting time.
You may assume given matrix equations are well defined (i.e. the matrix sizes are compatible).
If your final answer is short, put a BOX around it.
1. (24 points) Following the earlier command
>> Y = [1 2 3 4 5; 2 4 6 8 10; 3 6 9 12 15; 4 8 12 37 5; 5 10 15 5 69]
the MATLAB command
>> rref(Y)
produces the display
12300
00010
00001
00000
00000
(a) [8 pts] Write down a basis for the row space of Y.
(b) [8 pts] Write down a basis for the column space of Y.
(c) [8 pts] Write down a basis for the null space of Y.
2. (20 points) Dene S={xR3:x10, x20, x30 and x1+x2+x31}and
A=
1 2 2
302
541
. The volume of Sis 1/6. What is the volume of {Ax :xS}?
3. (20 points) Find the parameter values for β0,β1such that the line y=β0+β1xgives the
least squares best approximation to the following data points (xi, yi): (0,2),(2,0),(3,3),(5,6).
4. (25 points) Let Abe the matrix
1 4 5
0 2 6
0 0 3
. Find a matrix Uand a diagonal matrix
Dsuch that U1AU =D. (You do not have to compute U1.)
5. (30 points; 10 points for each part)
(a) Let Tbe the linear transformation from R2to R2which reects a point through
the line x2= 3x1. Find the matrix Asuch that T(x) = Ax for every xin R2.
(b) Let Bbe a matrix with characteristic polynomial χB(t) = t2+2t+ 1.
Find an orthogonal matrix Cwhich is similar to B.
(c) For the matrix Cabove, give a geometric description of the linear transformation
Sdened by the rule S(x) = Cx.
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