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(MATH113)113 finall99fall.pdf
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Math113 L2 Linear Algebra: Final Examination
Dept of Math, HKUST, Fall 1999
Name: Tutor:
ID No. Section:
Problem No. 1 (30 pts) No. 2 (52 pts) No. 3 (28 pts) Total (110 pts)
Score
3
2
3
2
.210 123
1. (5.6. 30 pts) LetA.
64
0.1 0
75
and Q.
64
012
75
.
0 1.2 001
(a)
Find all eigenvalues of A.
(b)
Find indep endent eigenvectors of A corresp onding each eigenvalue.
(c)
Compute the matrix A .
.1 365
(d)
Find all eigenvalues of Q A Q.
.1 365
(e)
Find indep endent eigenvectors of Q A Q for each eigenvalue.
89
Solution: (a) . . .1; .2.
3
2
1
64
75
. For . . .2, we have indep endent
(b) For . . .1, we have the indep endent eigenvector v.
1
1
1
3
2
3
2
10
64
0
75
andv.
3
64
0
75
eigenvectors v.
2
.
01
3
2
3
2
3
2
110 .100 010
64
100
75
.1
. ThenP AP .
64
0.2 0
75
.1
.P .
64
1 .1 0
75
89
. Thus A .
(c) SetP .
101 0 0.2 0.11
3
2
3
2
89
89 89
.100 .22.10
64
0.2 0
75
.1
P.
64
0 .1 0
75
.
P
89 89
0 0.2 0 2 .1.2
365
(d)...1;.2 .
.1 365.1.1 365
(e) If v is an eigenvector of A corresp onding tothe eingenvalue ., then Q A QQ v . Q A v .
.1 365 365.1.1 365 365
Q . v . . Q v. Thus Q v is an eigenvector of A corrp esp onding to the eigenvalue . .
3
2
3
2
100 0
.1
Therefore Q .
64
.2 10
75
64
.1
75
. For
.1
. For . . .1, the indep endent eigenvector u. Q v.
1
1
1.21 1
2
3
2
11
64
0
64
.2
.
365.1
.1
. . .2 , the indep endent eigenvectors are u. Q v.
andu.Q v.
2
2
3
3
01
1
2. (6.7+ 10 . 52 pts) Let v;v;v;v;vbe column vectors of the following matrix
12345
3
2
1010 1
A.
64
0100 1
75
:
0 0 1 0 .2
(a)
Find a basis for Nul A.
(b)
Find Spanfv;v;v;v;vg.
(c)
Find an orthonormal basis for Row A.
(d)
Find the orthogonal pro jection from R to Nul A and its matrix.
(e)
Find the orthogonal pro jection from R to Row A and its matrix.
(f)
Find the distances from the point (5;0;0;1; 0) to the subspaces NulA and Row A respectively.
(g)
(Bonus) Find an upper triangular 3 . 3 matrix R and 5 . 3 matrix Q such that the column vectors
12345
5
5
T
of Q are orthonormal and QR . A .
2
3. Part A: Circle one and only one answer for the following problems. (5 . 3 . 15 pts)
(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 n . n matrix. Which one is incorrect.
i. If 0 is an eigenvalue of A, then A is not invertible.
ii. If 0 is not an eig