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(MATH113)fall_final_98.pdf
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Math 113 Final Exam
Dec 14, 1998
Your Name

Student Number
Section Number

1.
Do not look at your book and notes. For more space, write on the opposite side.

2.
Do not use calculator. If your answer is too complicated, you must have made a mistake.

3.
Show all your work. Cross o. (instead of erase) the undesired part.

4.
Provide all the details. Your reason counts most of the points.


Number Score
1
2
3
4
5
Total

(1) (25 points)
Let
. .
0 0 1
A = . . 0 .10 . . .
1 0 0
1) (20 points) Diagonalize A;

2) (5 points) Use 1) to compute A5 +6A3 . 7A + I.
(2) (25 points)
1
Let W be the span of the following four vectors:

1 . . 1 . . 1 . . 1 . .
u1 = . . . . 0 0 . . . . , u2 = . . . . 1 0 . . . . , u3 = . . . . 1 1 . . . . , u4 = . . . . 2 3 . . . . .
0 0 .1 .3

1) (5 points) Check that
.. .. ..
10 0
.. .. ..
. 0 .. 1 .. 0 .
v1 = .. , v2 = .. , v3 = ..
. 0 .. 0 .. 1 . 00 .1
form an orthogonal basis for W .
2) (5 points) Let T : R4 R4 be the linear map de.ned as

1
T (x)=(v1 x)v1 +(v2 x)v2 +(v3 x)v3.
2
Here is the dot product. Find the standard matrix A for T . 3)(3points) What are the eigenvalues of A. 4) ( 10 points) Diagonalize A. 5) ( 2points) Compute 3A1997 . 7A1998 +10A1999 . 6A2000 .
(3)
(15 points, 3 points each) Let x be a unit vector in R10. Let A = xxT , i.e., the matrix product of x with xT . 1) Describe the column space of A. 2) Describe the null space of A. 3) What is the rank of A? Explain. 4) What are the eigenvalues of A? Explain. 5) Is A diagonalizable? Explain.

(4)
(15 points, 5 points each) 1) Find an example of 3 3 matrix of rank 2such that its square is equal to itself. 2) Find an example of 2 2matrices A and B, such that A and B have the same eigenvalues, both A and B are not diagonalizable, but AB is not

diagonalizable. 3) Find an example of 3 3 matrix A of rank 2such that A3 . 3A2 +2A =0.

(5)
(20 points, 1 point each) True or False (no reason needed) 1) If A isa5 3 matrix then its columns must be linearly dependent; 2)If A isa3 5 matrix then its rank must be 3 or less; 3) If Amnx = y has solution for any y in Rm, then m n; 4) If all eigenvalues of A are 1, then A is the identity matrix; 5) If A is similar to B, then A is diagonalizable if and only if B is diagonalizable; 6) If A is similar to B, then det A = det B; 7) If A is similar to B, then rankA = rankB; 8) Similar matrices have the same set of eigenvalues;


9) Similar matrices have the same set of eigenspaces;
10) If A is diagonalizable and all eigenvalues of A are 1, then A is the identity matrix.

Number Answer Number Answer Number Answer Number Answer
1 4 7 10
2 5 8
3 6 9

Answer
(1) det(A . I)=(1 . )(2 . 1). Therefore we
obtain =1, or .1 and the
corresponding eigenspaces (your answer may di.er by multiplying a number)

..