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Unformatted text preview: $‘E (# 171 E) Pagelof 5 it it} “F 3: j: 5% “fpﬁtiiﬁ?
The Chinese University of Hong Kong :0 DA 5 IO 01L tatﬁij‘éﬁaﬁ a Course Examination 2_ncl Term, 200i  2002 7H E £673 iii/Si Z #531
Course Code & Title : Faﬁ IJ‘ """"""""""""""""""""""""""""" " Time allowed : _________________________________ _» minutes same """"""" "I? """""" "
Student ID. No. Suppose that the transition matrix from X to Y is (a) (8 points) Compute the transition matrix from Y to X. (b) (8 points) Find X. (c) (4 points) Find the coordinate vector of z with respect to X, Where Course Code #5} E I a :_ E (alt i E) Page 2 of 5 2. (a) (6 points) Apply the Gram—Schmidt process to 1 0 0
u 1 u 2 1
: , = , U ——
1 0 2 1 3 0
2 2 2
(b) (6 points) Let
1 O 0 1
1 2 1 1
A = and b =
0 1 0 2
2 2 2 1 Show that the system Am 2 b is inconsistent.
(c) (8 points) Find the QR—factorization of A. (d) (3 points) Use 2(0) and the 4—vector b in 2(b), solve the least squares problem Arr: = b.
3. Let
2 2 —2
A = 2 5 —4
—2 —4 5 (a) (4 points) Find the coefﬁcients of the characteristic polynomial of A. (b) (12 points) Find all eigenvalues and corresponding eigenvectors of A? (c) (4 points) Is A diagonalizable? Explain. And, if so, exhibit an invertible
matrix P and a diagonal matrix D such that P“1AP = D. ,. Wm. w. “\NWW Course Code 7?} H 3,37% i .... ..................................................... .. a _=_ E (at— i E) Page 3 of 5 (a) (3 points) Let [3 be a real number and lﬁﬁ2
A: ﬂﬁ21
ﬁ2lﬂ Explain how the rank of A depends on ﬂ. (b) (3 points) Given V = R4, and 1131 ~ 2332
271,332 E R
~2$2 331 + 4332 Find UL. (C) (3 points) Suppose that B is similar to A. Show that if A is nonsingular, then
B is nonsingular and moreover, B‘1 is similar to A‘l. (d) (3 points) Let A be an n X n matrix, and let A be an eigenvalue of A. Show
that if 04 is any scalar, then A + Oz is an eigenvalue of A + 0:1”. Course Code #1 E 3,33% 2 % V—‘Jﬁeéi E) Page40f5 (a) (4 points) Let u be a vector in R" of length 1. Show that the matrix H=In —2'u,uT is symmetric and orthogonal. (b) Use 5(a), let 1) be any vector in IR” and 'w = Hv — v. i. (2 points) Show that w is a scalar multiple of u. 1
ii. (2 points) Show that 'v + 5w is orthogonal to u. 1
iii. (2 points) Draw and add the vectors w and v+§w to the picture as shown in Figure 1 on your examination paper. Note that H 'v is the reﬂection of
v in a plane perpendicular to u. V Figure 1: c.f., Question 5(b)— iii. Course Code £5} E sea}: I ﬁi§(#—£§) PageSofS 6. Let M be the vector space of 2 X 2 matrices, and let L : M ~—> 732 be the linear
transformation deﬁned by L([a =(a—b+c—4d)+(b+c+3d)m+(a+20—d)x2,
c
Where a,b,c, and d6 R. (a) (3 points) Find the matrix of L with respect to the natural (or standard) bases
B and C', for M and 732, respectively. (b) (3 points) Given an algebraic speciﬁcation for the range of L and use the
speciﬁcation to obtain a basis S for the range of L. (c) (3 points) For each polynomial r(x) in 8, ﬁnd a matrix A in M such that Let Bl denote the set of matrices found. (d) (3 points) Find a basis, B2, for the null space of L. (e) (3 points) Show that B1 U B2 is a basis for M. N The End of Examination N ...
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This note was uploaded on 03/03/2012 for the course MATH 2510 taught by Professor Jeff during the Fall '10 term at CUHK.
 Fall '10
 jeff
 Math, Linear Algebra, Algebra

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