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Math 111 Final Exam
June 1, 1994
Your Name
Student Number


Section Number

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

2.
Show all your work. Cross o. (instead of erase) the undesired part.

3.
Provide all the details. Your reason counts most of the points.


Number Score
1
2
3
4
5
6
7
8
Total

1

(1) (10 points) Consider

A =

1 10 5
0 .40

. .
5 10 1
..
1) Find the eigenvalues of A; 2) Find a basis of eigenvectors of A.
(2) (15 points) Consider vector space W spanned by

2

2

u1 =
,

u3 =
,

.

0

..
. ..
. ..
. ..
. ..
.
1

0

0

1) Show that u1,u2,u3 is a basis of W ;
2) Use Gram-Schmidt process to produce an orthogonal basis of W ;
3) Find the orthogonal projection of the vector (12, 0, 0, 0) onto W .

(3) (10 points) Consider the matrix

.

..
.
..
.
102
228

1) Find a basis for ColA; 2) Find a basis for NulA; 3) Find the rank of A.
(4) (10 points) Find numbers a, b, c, such that

. ..
. ,
2/3 .2/3
c
..
1/32/3 a

2/31/3 b

is an orthogonal matrix. Then .nd U.1 .
(5) (15 points) Consider

. 1 .21
6

A = ,u =

0

.

0

1) Find the distance from u to NulA; 2) Find the distance from u to (NulA) .
(6) (15 points) Find all the vectors

1

1

0

x =

,

,

1

0

x3
..
.
x4
..
. ..
. .
. ..
. ..
. ..
.
1

1

such that

217 2

x =0.
.12 .6 .1
(7) (12 points) 1) Find an example of 3 3 noninvertible matrices A and B, such that A + B is invertible; 2) Find an example of two bases {u1,u2,u3} and {w1,w2,w3}, such that {u1 +w1,u2 +
w2,u3 + w3} is not a basis;
3) Find an example of invertible 2 2 matrix that is not an orthogonal matrix;
4) Find an example of 22 matrices A and B, such that they have the same eigenvalues
but A is diagonalizable and B is not diagonalizable;
(8) (13 points) True or False (no reason needed) 1) If v1,v2,v3,v4 span a 4-dimensional vector space V , then {v1,v2,v3,v4} is a basis of
V ;
2) If all eigenvalues of A are 0, then A = 0;
3) If an invertible matrix A is diagonalizable, then A.1 is also diagonalizable;
4) If v is an eigenvector of A, then v is an eigenvector of A2;
5) If v is an eigenvector of A, then v is an eigenvector of AT ;
6) If u and v are eigenvectors of A, then u + v is also an eigenvector of A;
7) If {u, v, w} is an orthogonal basis of R3, then {2u, .3v, 5w} is also an orthogonal basis;
8) If u v, v w, then u w;
9) If u W and u W , then u = 0;
10) If the orthogonal projection of y onto W is .y, then the orthogonal projection of 10y onto W is 10.y;
11) If column vectors of a matrix is orthogonal, then the row vectors of the matrix is
also orthogonal; 12) If U and V are orthogonal matrices, then U + V is an orthogonal matrix; 13) If U and V are orthogonal matrices, then UV .1 is an orthogonal matrix.