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Unformatted text preview: g”? (é? 3 E) Pagelof 3 7% 5% ‘17 3c 2: $1 “Fiﬁ: $1“?
The Chinese University of Hong Kong Liv2?» eaﬁgeas E Area
Course Examination 2_ncl Term, 2009  2010 7H a as %&Z iii}
Course Code & Title : MAT2310C&D LINEAR ALGEBRA AND APPLICATIONS Ears """"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" " Time allowed 33’ i .453 55%
Student ID. No. SHOW ALL NECESSARY WORK TO GET CREDIT FOR SOLUTIONS
(NOTE: THERE ARE 100 POINTS TOTAL ON THIS EXAMINATION) . (10 points) Suppose that K 2 {W1, W2, W3, W4} is a linearly independent set of vectors in a
vector space V. Find the value(s) of t, if any when F : {X11x2)x3ax4} is also linearly independent, where
x1 2 W1 + th, x2 = W2 + tW3, X3 2 W3 + tW4, X4 = W4 + twl.
2. Answer each question separately. Show work to justify all answers. (a) (8 points) Let J = {w1,w2,W3} and K = {V1,V2,V3} be ordered bases for R3, where Determine J, if any. T
(b) (2 points) Let W = span{v1,V3} in R3. Write the vector v = [ 1 2 3 ] as W + u
with w in W and u in Wt. 1 1 (e) Let S = 0 , —1 be a basis for subspace W of the Euclidean space R3.
—2 0 T
Leth=[5 —3 —4] beinW. i. (2 points) Using the GramSchmidt process, transform 3 into an orthonormal basis
T for W. ii. (2 points) Find the length of h by using the coordinate vector of h with respect
to T. Course Code as as; : a 2 a (éi— 3 a) Page 2 of 3 (a) (2 points) Show that S is an orthogonal set.
(b) (4 points) Determine an orthonormal set T so that span 8 = span T. (c) (2 points) Find a nonzero vector u4 orthogonal to S and show that {1117 112, 113,114}
is a basis for R4 (d) Consider Ax = b, where A=[u1 u2 u3] and b: i. (2 points) Is b belong to the column space of A? Justify your answer. ii. (5 points) Compute the (QRfactorization of A. . >
) What can you say about QTQ? Justify your answer. ' . (4 points Use ii. and iii., solve for x. 4. Given that
2 1
A = 1 'y
7 O (a (4 points) Assume that rank A = 3. Find the value(s) of ’y, if any. )
(b) (6 points) Use 4(a), ﬁnd a basis for the null space of A and the nullity of A.
) (c (5 points) Assume that 'y = 1, show that the null space of AT is the orthogonal
complement of the column space of A. 5. (a) (8 points) Show that
(p, q) is an inner product on 732. (b) (2 points) In ’Pg, let p(m) = —2 + :L‘ + (c2 and q(m) = 1 —— 22: + 2502. Using the inner
product given in 5(a), ﬁnd the cosine of the angle 9 between p(:c) and q( (C) (2 points) Is {1, 33, 932} an orthogonal basis for 732, with the inner product deﬁned in
5(a)? Show work to justify your answer. (d) (2 points) Is the function deﬁned in 5(a) an inner product for 733? Show work to
justify your answer. Course Code HE éﬁat’lﬁ 1 a 3 E (;+T 3 E) Page 3 of 3 6. Given that the eigenvalues of A are 12 and 3 (multiplicity = 2), where 7 4 ~1
A: 4 7 —1
—4 —4 k (a) (2 points) Write down the characteristic polynomial of A.
(b) (2 points) Find the value(s) of k, if any. (C) (10 points) Find a basis and a dimension for each eigenspace associated with each corresponding eigenvalue, if any. 7. Answer each question separately. Show work to justify all answers. (a) Are the following statements true or false? i. (2 points) Let W = {V1,V2, V3} be a set of vectors in R3. If none of the vectors in W is a scalar multiple of any of the others, then S’ is linearly independent. ii. (2 points) The vector spaces R4 and 733 are isomorphic. (b) (2 points) Suppose A is an 3 X 3 square matrix. Let det(A + I3) = O, det(A + 213) = 0, det(A + 3I3) = 0. Fmddac4+4ny (c) (2 points) Suppose A = aaT + bbT, where a and b are in R3. Show that rank A S 2. (d) Let A represent an n X n matrix such that ATA = A2. i. (2 points) Show that naA—A%WA—AU)=0 ii. (2 points) Show that A is symmetric. N End of the Examination N ...
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 Fall '10
 jeff
 Math, Linear Algebra, Algebra

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