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**preview**shows half of the first page. to view the full**2 pages of the document.**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) Deﬁne S={x∈R3:x1≥0, x2≥0, x3≥0 and x1+x2+x3≤1}and

A=

−1 2 2

302

5−4−1

. The volume of Sis 1/6. What is the volume of {Ax :x∈S}?

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 U−1AU =D. (You do not have to compute U−1.)

5. (30 points; 10 points for each part)

(a) Let Tbe the linear transformation from R2to R2which reﬂects 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

Sdeﬁned by the rule S(x) = Cx.

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