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(math113)[2004](f)final~PPSpider^_10441.pdf
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Math113: Final Exam, Fall 2004
Name: Tutors Name:
ID No. Section:
Problem 1 (15) 2 (15) 3 (15) 4 (10) 5 (15) 6 (5) 7 (15) 8 (10) Total (100)
Score
1. (15 pts)
(a) Solve the system of linear equations:
x3 +2x4 +3x5 =0
x1 +2x2 +2x3 +2x4 +3x5 =0
(1)
2x1 +4x2 +x3 .2x4 .3x5 =0
2x1 +4x2 +3x3 +2x4 +3x5 =0
(b)
Find a basis for the orthogonal complement of the solution space of the linear system (1) in R5 .
(c)
Is there any linear transformation T : R3 R5 such that the range of T is the solution space of the linear system (1)? If yes, .nd the standard matrix of such a linear transformation T .
..
. .
.
2. (15 pts) Let P3 be the vector space of all polynomials in one variable t of degree at most 3. Note that B = {1, t, t2,t3} is a basis of P3. Let
f1(t) = 1+ t +2t2 +3t3 , f2(t) = 2+2t +4t2 +6t3 , f3(t) = 1+4t2 +6t3 , f4(t) = 3+5t +2t2 +3t3 , f5(t) = 2+3t +3t2 +3t3 .
(a)
Find the coordinate vectors for f1(t),f2(t),f3(t),f4(t),f5(t) relative to the basis B respectively.
(b)
Find a basis for Span{f1,f2,f3,f4,f5} from the set {f1,f2,f3,f4,f5} by row operations.
(c)
Determine the dimension of Span{f1,f2,f3,f4,f5}.
. ..
3. (15 pts) Let A =
., where a is an unspeci.ed parameter.
(a)
Calculate det A.
(b)
Find all values for a so that the the matrix A has zero as an eigenvalue.
(c)
Find all values for a so that the rank of A is 3.
..
12 21
4. (10 pts) Let B = {v1, v2, v3} and C = {w1, w2, w3} be two bases for a subspace W of R4, where
v1
=
..
.
1
1
0
0
..
.
, v2
=
..
.
0
1
1
0
..
.
, v3
=
..
.
0
2
1
.1
..
.
;
w1
=
..
.
1
2
2
1
..
.
, w2
=
..
.
2
1
.1
0
..
.
, w1
=
..
.
0
.1
1
2
..
.
.
....
(b)
(Not for students in Prof. Qians Lecture.) If a vector w in W has the coordinate vector
(a)
Let v be a vector in W whose coordinate vector relative to the basis C is
. ..
2
3
1
..
Find the coordinate
. relative
vector of v relative to the basis B.
. ..
a
b
c
to the basis C, .nd its coordinate vector relative to the basis B in terms of a, b, c.
5. (15 pts) Let A =
200
135
002
. ..
..
(a)
Is A diagonalizable? If yes, .nd a matrix P such that P .1AP is a diagonal matrix.
(b)
Compute the matrix A100 .
(c)
Let Q be an invertible 3 3 matrix. Is the matrix B = QAQ.1 diagonalizable? If yes, .nd a matrix U such that U.1BU is a diagonal matrix.
..
6. (5 pts) Let A be a 3 3 matrix having distinct eigenvalues 1, .1, 0. Show that A3 = A.
7. (15 pts)
(a) Let W be a subspace of R4 de.ned by the linear system Ax = 0, where
A =
..
.
1 .1 .11
1 .1 .11
1 .1 .11
1 .1 .11
..
.
.
Find the shortest distance from the point (1, 2, 2, .5) to W .
(b) Find the standard matrix of the orthogonal projection ProjW : R4 R4 . 8. (10