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(MATH113)113 finall98spring.pdf
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Math113 Linear Algebra: Final Examination
Dept of Math, HKUST, Spring 1998
Name: Tutor:
ID No. Section:
Problem No. 1 (100 pts) No. 2 (90 pts) No. 3 (90 pts) No. 4 (70 pts) No. 5 (150 pts) Total (500 pts)
Score
44
1. Consider the linear transformation T : R .! R given by
3
2
1
3
2
0
xx.x+x
1 123
T
6664
BBB@
x
x
C
7775
.
6664
7775
:
x+2x.x
C
2
1
3
4
C
A
2x.x+3x.x
3
1
2
3
4
xx+x+3x.2x
4 1234
(a)
Write the standard matrix A of the linear transformation T . (30 points)
(b)
Find bases for Nul A, Row A and Col A. (30 points)
3
2
1
3
2
0
3
2
xx1
11
(c) Find all vectors
6664
x
x
7775
such that T
6664
BBB@
x
x
C
7775
.
6664
1
2
7775
. (40 points)
C
2
2
C
A
3
3
xx1
44
#
"
#
"
#
"
14 0.2 1.2
2. LetA. ; ;and .
32 .20 81
.1
(a) Which one is orthogonally diagonalizable. If so, .nd an orthogonal matrix Q such that Q AQ is
diagonal. (30 points)
.1
(b) Which one is diagonalizable. If so, .nd a matrix S such that S AS is diagonal. (30 points)
#
"
a .b
.1
(c) Which one is not diagonalizable. If so, .nd a matrix P such that P AP is of the form .
ba
(30 points)
3
2
3. Let A.
64
211
121
75
.
112
(a)
Is A diagonalizable. If yes, diagonalize A. (30 points)
(b)
Is A orthogonally diagonalizable. If yes, orthogonally diagonalize A. (30 points)
(c)
Can A be written as A . B . If yes, .nd such a matrix B. (30 points)
2
3
2
4. Is the set of vectors perp endicular to
6664
1
.1
1
.1
7775
4
in R a subspace. If yes, .nd an orthogonal basis of the
subspace. (70 points)
5. Part A: Circle one and only one answer for the following problems. (6 . 20 . 120 points)
(a)
Let A be an m . n matrix. Which one is correct.
i. dim Row A . dim Nul A;
ii. dim Row A . dim Nul A;
iii. dim Row A . dim Nul A;
iv.
dim Row A . dim Col A;
v.
dim Row A . dim Col A;
vi. dim Row A . dim Col A;
vii. dim Nul A . dim Col A;
viii. dim Nul A . dim Col A;
ix. dim Nul A . dim Col A.
(b)
Let A be an m . n matrix. Which one is incorrect.
1
i. If A is non-singular, then Ax . 0 has only one solution;
ii. A is singular, then Ax . 0 has in.nitely many solutions;
m
iii. If A is non-singular, then Ax . b has only one solution for any b of R ;
m
iv.
A is singular, then Ax . b has in.nitely many solutions for any b of R ;
v.
None of the above.
(c)
Let A be a square matrix. Which one of the following is correct.
i. Row A is perp endicular to Col A;
ii. Row A is perp endicular to Nul A;
iii. Col A is perp endicular to Nul A;
iv. None of these.
(d)
Let A be an m . n matrix. Which one of the following is true.
n
i. If ~v; . . . ; ~vare linearly indep endent vectors in R , then A~v; . . . ; A~vare l